help-17

[urban edge]

so the first thing we want is to find the bounds of our integral

[cunning slate]

how would we do that?

[urban edge]

since we want to find the area between two curves, we need to fins when they intersect

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[vast shale]

2.38

i dont quite understand

"if the expression is the same for all values of x"

[surreal basin]

consider something like 5x - 5x + 7

• this is an expresion
• no matter what value x has, the expression is 7
• so, this expression is the same for all values of x

are you with me so far

[vast shale]

ohh

[vast shale]

so 3(x+7)-(2x+9) = 3x + 21 - 2x - 9

?

[surreal basin]

3x + 21 - 2x - 9
3x - 2x + 21 - 9
x + 12

x + 12 changes for all values of x
1 + 12 is not the same thing as 2 + 12
so no, you did not successfully read the problem

this is a blank, you are supposed to put a number inside the blank

[surreal basin]

you have to put a number inside the blank so that something like this will happen

[surreal basin]

if you put nothing in the blank, it is incorrect

@vast shale to start, do you know why 5x - 5x + 7 always remains the same for all values of x?

[vast shale]

hmm

[vast shale]

because of constant 7?

[surreal basin]

not really

what about that 5x - 5x?

[vast shale]

because x is unknown and therefore it remains?

[surreal basin]

think about 0 times x

what is 0 times 5?

[vast shale]

0

[surreal basin]

what is 0 times 100?

[vast shale]

0

[surreal basin]

what is 0 times anything?

[vast shale]

0

[surreal basin]

what is 0 times x?

[vast shale]

0

[surreal basin]

correct

so 0x is just 0

that means the x didnt change anything

[surreal basin]

so this reason doesnt wokr

x is unknown, but 0x is very much known, its always 0

it doesnt matter what x would be for 0x to be 0, right

[vast shale]

yep

[surreal basin]

now lets try 5x - 5x

whats 5 * 5 - 5 * 5?

[vast shale]

25-25

0

[surreal basin]

whats 5 * 7 - 5 * 7?

[vast shale]

35-35 = 0

[surreal basin]

whats 5 * 100 - 5 * 100?

[vast shale]

500-500 = 0

[surreal basin]

do you need to calculate 5 * 100 to see that 5 * 100 - 5 * 100 is 0?

[vast shale]

nope

because they have the same factor we can just minus out 5

[surreal basin]

you mean to do 5x - 5x = (5 - 5)x?

[vast shale]

yea

[surreal basin]

thats correct

and so you get 0x as before

which we know is 0

[vast shale]

uhuh

[surreal basin]

so 5x - 5x is always 0, right?

[vast shale]

yeap

[surreal basin]

now lets look at this

you can put any number in this blank

but youre looking to put in a particular number

its a fraction

you need to put a fraction in this blank

so that the xes subtract each other out

just like they would with 5x - 5x

3x + 21 - ??? * (2x + 9)

do you have any ideas on what would work in that ???

[vast shale]

so i need to form an equation?

[surreal basin]

cmon

[vast shale]

3/2?

[surreal basin]

thats the one

thats the answer

[vast shale]

but how

[surreal basin]

you guessed 3/2

its correct

[vast shale]

is there like a systematic approach?

[surreal basin]

no

you

1. read the problem
2. see how it works
3. figure it out

[vast shale]

damn..

[surreal basin]

youre not going to memorize the solution to every problem out there

this is one of them

thats why its in the challenge problems

youll just need to figure out that its coming

[vast shale]

did you tried AOPS books?

[surreal basin]

I dont use them, but if you need them thats up to you

[vast shale]

i see

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[fickle frost]

Can someone help me with this question like where did the underlined part come from? is it like multiplying x^2+1 to all values to cancel out the denominator?

[hazy bone]

Hey

Idk how this works, where do I ask the question?

[fickle frost]

Uhm in the available helps

[hazy bone]

But you are already asking for help in a question here, can I still ask mine over that?

[fickle frost]

Not really

[hazy bone]

Alright, I will find a empty one ig

[deft mortar]

@fickle frost

hi

[fickle frost]

hi

[deft mortar]

that multiplies fractions

remind, every number is always divided by 1

so ln(xx+1) is also = ln(xx+1)/1

then multiply extremes and middle

to get only one fraction

[fickle frost]

oh okay thanks

[deft mortar]

okay

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[formal obsidian]

I'm very confused

here's my work:

sps the velocity of the airplane is u = <0,300>, and the velocity of the wind is v = <0,30>

a=u+v=<0,330>

a=330cos90i + 330sin90j

for the airplane should arrive at a point due west, the velocity must be <-330,0>

330cos(theta)=-330
arccos(-1)=180=theta

so now my answer is to head -90 degrees N?

which is just wrong tbh

okay update: i thought that maybe i should think of it as the plane has already moved towards west with a velocity of <-300,0> then the wind should be blowing it to it's side which produces a vector of the plane's velocity and the wind's getting added

now the plane have to adjust for that angle?

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[formal obsidian]

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[minor carbon]

is P^-1 correct?

its wrong no?

[tired gate]

It's wrong yeah, the last row times the first column doesn't give 0

[half imp]

?

Shouldn't it give 1

oh last row

[tired gate]

Mb on the late edit haha

[half imp]

,w [[1,1,0],[1,1,1],[0,0,-1]]^-1

[twin meteor]

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[minor carbon]

thanks!

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[half imp]

np

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[wary mantle]

Now what?

[wraith venture]

don't you have a theorem about alternating sums?

[wary mantle]

I have this

[wraith venture]

yes

hence a very simple bound on the error

[wary mantle]

Oh

[wary mantle]

So we set $\frac{(k+1)}{2^{k+1}} = 10^{-3}$, right?

[twin meteor]

Kepe

[wary mantle]

Thanks

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[wary mantle]

One more thing:

[wary mantle]

It says a_k >= 0 here

But we don't need that?

Do we?

[wraith venture]

it has to be alternating

[wary mantle]

Well what if a_k < 0

Then it can alternate too?

[wraith venture]

so long as it's of constant sign

[wary mantle]

Hm, why does it say a_k >= 0 then

Probably a mistake

[wraith venture]

wlog you can assume so

just consider - your series otherwise

[wary mantle]

Alright

Thanks

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[toxic shuttle]

how do i do this question? im confused af

[toxic shuttle]

and quotient obtained while dividing, does it mean i directly divide equation p(x)/g(x) like that?

i think option 3 is correct but i dont understand option 1 and 2

[loud walrus]

When x goes to infinity in q(x)

What do you get?

[oak magnet]

The limit of a polynomial is the limit of the dominant coefficient

[loud walrus]

So what would you get?

[toxic shuttle]

what is dominant coefficient?

[toxic shuttle]

so i need to transpose the value of x to infinity?

[oak magnet]

Did i say something wrong ?

[toxic shuttle]

i honestly have no idea

[toxic shuttle]

like consider the value of x as infinity?

if yes, then the whole equation will turn infinity

[outer warren]

yes, dominant coef is not the correct terminology

[oak magnet]

What should i say, not a great english speaker, i translate litteraly

[loud walrus]

You sure?

[toxic shuttle]

how do i calculate turning points though? like i plot it on graph and see how many turns the graph will take?

[outer warren]

leading term,
not coefficient as that only refers to the constant component

[toxic shuttle]

not really but like infinity raised to power something will still be infinity right?

i mean like at the end it will be infinity - 2

[loud walrus]

But it is -x^5

[toxic shuttle]

OH WAIT

[loud walrus]

-1(x^5)

[toxic shuttle]

so like it will be - infinity

and then the other will be + inifnity

[loud walrus]

Then, the first is true or false?

[toxic shuttle]

and both will cancel out

[loud walrus]

No

[toxic shuttle]

false

[loud walrus]

Read the exercise

Ok

The second sentence

Is true or false

[toxic shuttle]

false

[oak magnet]

Ok yeah i see the mistake ty

[loud walrus]

Why

[toxic shuttle]

because both of them will lead to - infinity

[loud walrus]

Read again

[toxic shuttle]

instead of +infinity as it seems in the options

[loud walrus]

P(x)

When it goes to -inf

[toxic shuttle]

wait no

[loud walrus]

-x^5

[toxic shuttle]

option 1 will be false, 2 will be true

[loud walrus]

Exactly

[toxic shuttle]

as the question says to take x = -infinity

damn

i was misreading it

[loud walrus]

I let u finish the other two by yourself

[toxic shuttle]

how do i calculate turning points?

and by option 4, i just divide p(x) by q (x)? @loud walrus

sorry for the ping

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[humble surge]

Hi there, a short combinatorics/probability question. Should be around highschool level (Ive forgotten a lot):
How can I calculate the chance c that
play over r rounds
with l as chance to lose
that I will lose t times in a row at least x times?

[humble surge]

My working till now was just setting t=3, x=1, r=200, l=0.3 just to test things. So for 3 rounds the chances are 0.3^3. For 4 rounds I have 2 possibilities of 0.3^3 occuring, so 0.3^3*2?

For 5 rounds it looks like it should be 0.3^3 * 3, which led me to believe that it should be (r-t+1)*l^t=c

This is obviously wrong, since it can exceed 1, although I expect a chance.

I hope the question is concise enough; can anyone nudge me in the right direction?

one of my problems is that all possible outcomes r^2 (since one can only win or lose) are not evenly distributed.

If anyone can help me to formulate the problem better (and with the right vocabulary) I am all ears!

<@&286206848099549185> anyone any idea?

[vocal sleet]

@humble surge Has your question been resolved?

[humble surge]

<@&286206848099549185> Does my question need reformulation?

[vocal sleet]

@humble surge Has your question been resolved?

[humble surge]

clarification: the loss-streaks can overlap. the chance to win/lose each round are independent of each other

[humble surge]

Probably I misjudged the difficulty of the problem, might this be university level?

[knotty lynx]

I think your formula is missing (1 -l)^(r-t) for the rounds where you win

[humble surge]

Im not sure I follow; why would I care about the rounds where I win?

I thought I need to find all combinations there are (1). Then find all combinations with at least t losses in a row (2). Then calculate the chance of these rounds (3). Add all the chances up (4). That would be the plan, but I fail to see how to express it mathematically

[vocal sleet]

@humble surge Has your question been resolved?

[knotty lynx]

Consider these values
t = 3, x = 1, r = 5, l = 0.3

There are 3 scenarios where we lose 3 in a row:

3. • • ---

'+' win
'-' loss

In the first scenario, we encounter 3 losses and then 2 wins
In the second scenario, it's one win, 3 losses, one win

For example, the probability of the second scenario occurring over 5 rounds is 0.7 x 0.3^3 x 0.7

In every scenario, there are 3 losses and 2 wins, all of which are independent of one another so the probability of one singular such scenario is always 0.7^2 x 0.3^3

We want to consider the probability of any of them occurring. Each one has the same probability so we can just multiply the probability of the number of scenarios. That is (r - t + 1) as you've noted

I realize you are likely aware of most of the stuff I mentioned later in this message. It's just there for completeness' sake

[humble surge]

Ah no worries! Thanks for helping and untying the knot in my brain 🙂

[knotty lynx]

Glad I could be of some help

[humble surge]

well, Im not 100% convinced yet. Because the scenario: ---+- has "only" a chance of 0.3^4*0.7 of occuring, right? We still want to include that as well

if it was only about losing 3 times and winning the rest you are certainly right

[knotty lynx]

Ah you're right! I completely missed that

[humble surge]

but youre right, order doesn't matter! Didn't see that clearly. Honestly I am thinking about closing the topic. I just guessed that it should be doable with high school lvl stochastics, but the more I think about it the more I think it might be uni-level stuff

[knotty lynx]

Idk what necessarily constitutes as high school maths but I think the solution won't be too ridiculous

[humble surge]

well, I leave it open then 🙂 but it sounds just like a math problem of high school where I fail to see the solution...anyway thanks for helping!

[dark kiln]

this is hopeless

[humble surge]

@dark kiln You mean to discuss it here further?

[dark kiln]

no sorry

[knotty lynx]

no problem. I'll probably have a look again tomorrow

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@humble surge Has your question been resolved?

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[proven nebula]

are 0.999... and 1 the same number?

[whole oasis]

yes

[proven nebula]

but why

[hard atlas]

the first question you should ask yourself is: what does the notation 0.99999.... even mean

[whole oasis]

Well, there are several proofs

e.g.

[plain minnow]

This is not a proof before you have a clear definition of what "..." means in this context and that a consequence of this definition is that 10 · 0.999... = 9.999... and 9.999... = 9 + 0.999...

[whole oasis]

Okay that's not an elementary proof, but let's call it an algebraic argument which "helps" to understand the process

[vocal sleet]

@proven nebula Has your question been resolved?

[proven nebula]

huh?

im confused

[urban iris]

think of number between 0.999… and 1

you can always find a real number between two distinct real numbers

eg : between 1 and 2 you can have a distinct real number by just averaging ie 1+2/2=1.5

between 1 and 1.5 you can do the same thing 1+1.5/2 =1.125

and you can continue

[proven nebula]

shouldnt there be a 0.0000....1?

infiniteismal?

[urban iris]

why?

[jade anvil]

Yes, it is infinitesimally close to 0

[urban iris]

no it isnt

[wary mantle]

Such a number doesn't exist

[urban iris]

there is NO number between 0.999999.. and 1

because that’s what 0.999… means

[wary mantle]

But yeah, it's probably the best to write down what $0.999\dots$ is even supposed to mean. [0.999\dots \coloneqq \sum_{k=1}^\infty 9 \cdot 10^{-k} = \sum_{k=1}^\infty \frac{9}{10^{k}} = 9 \cdot \sum_{k = 1}^\infty \frac{1}{10^k}] Now $\sum_{k = 1}^\infty \frac{1}{10^k} = \sum_{k = 1}^\infty \l(\frac{1}{10}\r)^k$ is a power series with $\l|\frac{1}{10}\r| < 1$ and thus we have that its value is [\frac{1}{1 - \frac{1}{10}} - 1= \frac{10}9 - 1 = \frac19] (where we need $-1$ because the formula only holds starting from $k = 0$), and thus [9 \cdot \sum_{k = 1}^\infty \frac{1}{10^k} = 9 \cdot \frac{1}{9} = 1.] That's one possible argumentation.

[urban iris]

you would need to use limits then

this gives off the idea that its “infinitesimally close to 1”

[proven nebula]

wha?

[plain minnow]

An infinite sum is a limit

[urban iris]

you can have no number between 0.99.. and 1

[proven nebula]

why

[urban iris]

think about what 0.99… means

[proven nebula]

1/3*3

i get that

but shouldnt it be a paradox

[urban iris]

dont use tht

[proven nebula]

oh okay

can u use limits cuz i dunno summation

[urban iris]

no need to use limits either

[wary mantle]

Well we can argue completely differently too

[proven nebula]

we can?

[urban iris]

i dont really like this since you need to first define what the infinite series “converging” to the number actually means

[cobalt crypt]

i dont really like this since you need to first define what the ... in 0.999... means and what real numbers are

[urban iris]

this gives off the misconception that there is 0.000000000..1 between 0.99.. and 1

[cobalt crypt]

maybe there is

[urban iris]

there isnt?

[cobalt crypt]

how would you know if you dont have a definition of the real numbers

[urban iris]

https://en.wikipedia.org/wiki/0.999...

[cobalt crypt]

you must have some mention of a properties of the real numbers

[cobalt crypt]

the fact that there is no number between 0.999... and 1 is a consequence of the archmedean property, and can fail in other number systems

so you need to know, as denascite began with, what 0.999... means

and what the real numbers are

[urban iris]

doesn’t 0.99… literally intuitively mean you can’t have a number between that and 1

[cobalt crypt]

you need to explain this concern

at least if you extend to the hyperreals, thats not a crazy thing to say

[urban iris]

so if there’s 0.000000..1 lying somewhere => 0.99… is terminating right?

[cobalt crypt]

no?

[urban iris]

why

[wary mantle]

Both could be not terminating, one could think

[cobalt crypt]

maybe i have a digit which is the (ω + 1)th digit

and it is 1

but all ω digits prior are 0

[urban iris]

that only makes sense when both are terminating no?

[cobalt crypt]

ω is infinite

but right after that, you have a 1

[urban iris]

can you just construct that? if you have 0.99… and 0.(w) digits and then 1 , theres always a one-one correspondence to every digit of 9 to 0. you placing a 1 at the end makes me think its terminating at a 1

whats after the 1?

[proven nebula]

if w+1 =1 then shouldnt w = 0?

(im going to ask a lot of dumb questions)

[urban iris]

in 0.000…1 whats after the 1?

[proven nebula]

and arent the negative numbers before 0 counted?

why are numbers prior w 0

[cobalt crypt]

dont worry about it

its not possible in the reals, but only because the real numbers has a precise definition

but im making the point that you need to know what the real numbers are and their properties to give a proper argument discounting the fact that there cannot be a 1 following infinitely many 0s in the decimal expansion of a number

[haughty swan]

<@&268886789983436800>

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[plush shoal]

what is f(c)

[plush shoal]

whats the c specifically

[tawny hollow]

c is presumably some fixed real number in this case

[peak matrix]

c is the point at which you are testing continuity

function f is continuous at point c, if those 3 conditions hold

[plush shoal]

its a bit confusing ngl

cuz why not just x

[peak matrix]

because x is already there, you cant use the same variable twice

and "c" looks more like a constant

e.g. function f is continuous at point 2, if:

1. f(2) exists
2. lim x->2 f(x) exists
3. lim x->2 f(x) = f(2)

c can be any number of your choice

and those 3 conditions then determine whether the function is continous at c, or not

[plush shoal]

i think i get it now

thanks

[peak matrix]

np

[plush shoal]

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[daring pumice]

can someone check if this is right

i never did math before

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[tawny grotto]

Hi how would I solve this?

[jade anvil]

Where are you stuck?

[tawny grotto]

I dont know where to begin to solve it

[vocal sleet]

@tawny grotto Has your question been resolved?

[tawny grotto]

Uh im not sure

ok so would 3 be 27?

[twin meteor]

faiyrose

[twin meteor]

faiyrose

[tawny grotto]

Yes

so we would do the same to 5 as well

and

it would be 27 and 125?

[vocal sleet]

@tawny grotto Has your question been resolved?

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[marsh raven]

could anyone explain this to me? i am horrible with word problems and im questioning if what ive done already is right 😭

[astral pilot]

let's say there were x child admissions

[marsh raven]

alright

[astral pilot]

so how many adult tickets were sold then?

[marsh raven]

2x?

[astral pilot]

yes

so each child ticket costs 6.40 right?

[marsh raven]

yeyes

[astral pilot]

so how much would x child tickets cost?

[marsh raven]

6.40x

[astral pilot]

yes

now, what would the cost of 2x adult tickets

[marsh raven]

uhhm

[astral pilot]

it would cost 2x* 9.60

because 1 ticket costs 9.60

[marsh raven]

ohh

[astral pilot]

6.40x + 19.20x = 640

ig you can do it from here

[marsh raven]

thank u so much

ahhhh thank u sosososo much that was really helpful🙏

[astral pilot]

u r wc

[vocal sleet]

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[odd olive]

Can anyone help

[astral pilot]

[odd olive]

I am asking🤨

I wanna solve solve this by considering x+....+x^n a geometric sequence of first term x and common ratio x

[peak matrix]

sure? You definitely can

though i actually think lhop would be much faster here

[odd olive]

It would yes

But I'm trying this method
For some reason I'm not getting the same answer idk what I'm doing wrong😭

[loud walrus]

n=1+1…(n times)

[signal pendant]

Show us your steps

[odd olive]

Okay I figured it out

[vocal sleet]

@odd olive Has your question been resolved?

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[lost ravine]

We need to solve the inequality and find the sum of the whole solutions. I've already solved such tasks, but this time the answer is wrong, I'm probably doing something wrong

[terse forum]

in 4th step where you messed up

actually its 3rd step

apply the absolute rule instead

solve for both x, 2x-3>-8 and 2x-3<8

[vast shale]

that part is right, i think

[outer warren]

what's the reasoning for your choice of values for the summation

[lost ravine]

Can you please explain in more detail?

[lost ravine]

arithmetic progression

[outer warren]

solving the inequality itself was fine

explain in detail your choice of values

[terse forum]

I recall when you have a problem such as |x| < a so you could do -a<x<a

for example, let's say we have |x - 3| < 9, simplify it -9 < x - 3 < 9 then gives us -6<x<12

[outer warren]

that was done in their first step

[terse forum]

I thought you could divide but I see they subtracted there

wait i see in decimal number nvm

[terse forum]

could you tell me where did you get 4.5 from?

[lost ravine]

The integers in the interval (-2.5;5.5) form an arithmetic progression with the first term -1.5 and the last term 4.5

[outer warren]

do you know what integers are?

[lost ravine]

yes

[outer warren]

is -1.5 an integer?

[lost ravine]

Well, it's a fractional number

[outer warren]

not what I asked

[lost ravine]

What did you mean by that?

[outer warren]

it's a yes or no question

[lost ravine]

no

[outer warren]

ok. there's your mistake

you're consider the sum of values that aren't even integers

[lost ravine]

Omg, what should I do now?

[outer warren]

list out the values that are actually integers

[lost ravine]

which are in the range -2.5;5.5 ?

[outer warren]

yes

[lost ravine]

-2;-1;0;1;2;3;4;5

[outer warren]

yeh

and you can use the sum formula for that if you really want
or just add them up normally

[lost ravine]

S=4+2/2*4=12 the correct answer is 12

[lost ravine]

thanks for the help

[outer warren]

why 4+2

end result is fine, but curious why/how you're plugging in values like that

[lost ravine]

I think I made a mistake.

[lost ravine]

I was wrong, the correct answer turned out to be accidental

[outer warren]

do you know what
a_1
a_n
n
represent in that sum formula

[lost ravine]

a1 is -1 ?

[outer warren]

no

in general, do you know what
a_1 of a sequence means

[lost ravine]

no

[outer warren]

the subscript here is an indication of the term number

a_1 represents the first term
a_2 the second term
etc...

[lost ravine]

the number of the member in the period?

[outer warren]

wdym in the period?

[lost ravine]

these are all numbers from -2 to 5

[outer warren]

uh poor wording but I suppose?

still not clear if you understand

[lost ravine]

No, it's not quite clear

[outer warren]

can you try telling me the value of
a_1 again

[lost ravine]

a1 is the ordinal number of the number that is in the period (-2;5)

[outer warren]

?

I'm expecting a single value with no extra padding attached

[lost ravine]

Okay, what formula did you have in mind?

[outer warren]

not a formula

just the definition

that you think was quite clear

[lost ravine]

and so, to find the sum of the whole solutions, do I just need to add up all the numbers from -2 to 5?

[outer warren]

careful with whole and integer as those are different

whole only refers to positive integers

and yes, I said that you could just add up the numbers normally that way

if you want to apply the sum formula, you'd need to understand it

[lost ravine]

can you show me this summation formula

[outer warren]

you wrote it down earlier

sum of the first $n$ terms of an AP
$$S_n = \frac n2 (a_1 + a_n)$$

[twin meteor]

ℝαμΩℕωⅤ

[lost ravine]

what is n and a1b, an ?

[outer warren]

n is the number of terms you're adding

a_1 as mentioned earlier is the first term

a_n is the nth term (which will be the last term)

[lost ravine]

S= 5/2(-2+5)

[outer warren]

no, n, the number of terms isn't 5

[lost ravine]

so, S= 8/2(-2+5)= 12

[outer warren]

yes

[lost ravine]

S= a1+an/2n=-2+5/28=12

[outer warren]

missing a lot of ()

[lost ravine]

what ?

[outer warren]

() are needed to clearly communicate order of operations in text

[lost ravine]

that was done in their first step

[outer warren]

for fractions, they're necessary to clearly indicate what's actually supposed to be in the numerator

and denominator

[outer warren]

the fraction line in the second line should be longer
but writing without () like that is fine due to the advantages of horizontal fraction lines

the issue I'm raising is that you don't have that benefit in text

[outer warren]

if I typed \verb|-2 + 5/2| would you interpret that as \ \
$-2 + \frac 52$ or $\frac{-2+5}{2}$

[twin meteor]

ℝαμΩℕωⅤ

[lost ravine]

1)-1/2 2) 3/2 ?

[surreal basin]

monul its a question

it says "would you interpret that as"

do you know what this phrase means

[lost ravine]

Sorry

[outer warren]

not what I asked

not asking you to calculate anything at all

[lost ravine]

I realized that these are two different tasks and the answer will be different, is that what you meant?

[outer warren]

not quite

for what I asked it's purely a question about interpretation of notation

[lost ravine]

omg, I get it, I would understand this as the first option

[outer warren]

yes

thus why I mentioned you would need () if you intended the second

[lost ravine]

yes I see

[lost ravine]

Thank you so much for your help

[vocal sleet]

@lost ravine Has your question been resolved?

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[finite hatch]

uh..

[finite hatch]

how do i approach this

[vocal sleet]

@finite hatch Has your question been resolved?

[vocal sleet]

@finite hatch Has your question been resolved?

[vocal sleet]

@finite hatch Has your question been resolved?

[blissful sentinel]

is this from the artofproblemsolving website haha, is there not an answer key

[finite hatch]

idk i got sent it

where can i find the solution

im freaking out it is 6 am

[blissful sentinel]

is this for a class?

[finite hatch]

uh idk

its to prepare for a math competition

[blissful sentinel]

why are you up doing this at 6 am

[finite hatch]

australian maths competition

[blissful sentinel]

😭

[finite hatch]

bc i need to get a certificate of distinction or high distinction

and my mum will buy me a car

[finite hatch]

do u know where i can find solution?

[blissful sentinel]

You might be able to find it on artofproblemsolving? Not sure how you would look up this specific question

[finite hatch]

😔😔

i ceebs

maybe i should give up

[blissful sentinel]

There's a whole range of other questions you could practice using though, if you wanted

[finite hatch]

oo ok thank u

good night :F

:D

.close

[vocal sleet]

Channel closed

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Use .reopen if this was a mistake.

[blissful sentinel]

I do think I figured it out though

[finite hatch]

.reopen

[vocal sleet]

✅

[finite hatch]

oooo

[blissful sentinel]

You just need to use the fact that
[\tan(2023x) = \frac{\sin(2023x)}{\cos(2023x)}]

[twin meteor]

biased_estimatERIC

[finite hatch]

i would not think of that 😭😭

[blissful sentinel]

This part is just the definition of tangent so far

Or at least one of them

Whenever you see tangent, it's helpful to write sine / cosine instead

And then you can use de Moivre's formula to calculate the numerator and denominator

[finite hatch]

ahhh ok thank u

what is moivres formula?

[blissful sentinel]

Maybe you should try some easier problems involving it

De Moivre's formula says that
[\cos(nx) + i\sin(nx) = (\cos x + i\sin x)^n]

[twin meteor]

biased_estimatERIC

[blissful sentinel]

Basically it gives you a fast way to calculate cos(nx) and sin(nx)

[finite hatch]

is there multiple ways to solve?

i dont think ive come across that

formula

[blissful sentinel]

This way is rather simple

It is a helpful thing to know if you are interested in solving trigonometry competition questions

If you know Euler's formula (e^{ix} = \cos x + i\sin x), then de Moivre's formula is an easy consequence of it

[twin meteor]

biased_estimatERIC

[blissful sentinel]

Once you plug this in and use the binomial theorem though, it just becomes a matter of simplifying things correctly, and you'll get the answer

[finite hatch]

wow...... to be honest i dont even know eulers formula

ive heard about it though

[blissful sentinel]

If you're not familiar with these formulae though, you might want to try some easier problems

[finite hatch]

in hidden figures the movie

[blissful sentinel]

Artofproblemsolving has a TON

[blissful sentinel]

hahaha good movie

Like here are a ton of practice problems from american math contests: https://artofproblemsolving.com/wiki/index.php/AMC_Problems_and_Solutions

[finite hatch]

yes i think i need that..

im in 10th what are these formulas 😔😭

[blissful sentinel]

they also have this thing alcumus that gives you practice problems that get harder and harder based on your level https://artofproblemsolving.com/alcumus/problem

[finite hatch]

ooo like tailored?

ok thank u so much :D

i hope i get my car 😔😔

[blissful sentinel]

yea

[blissful sentinel]

getting a car as a present for a math contest is absolutely wild

how rich are you lol

[finite hatch]

noooo not rich 😭😭

my mum just wants to encourage idk

so she can frame it

[blissful sentinel]

good luck!

[finite hatch]

ok thank u :D

.close

[vocal sleet]

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[lofty agate]

How do I prove this inequality? With z and w being complex numbers:

[vocal sleet]

@lofty agate Has your question been resolved?

[vocal sleet]

@lofty agate Has your question been resolved?

[normal heath]

If i didnt do a mistake, using z = r e^i phi and same for w solves it after simplifying a bunch, and using cos(...) <= 1

Also squaring everything can be helpful, and equivalent because both sides are positive reals

[lofty agate]

I've tried that, but... I didn't reach anything helpful

[normal heath]

Ok well heres my very crude working if you want to have a look...

z = x* = a e^id and w = v* = b e^ip

And all the lines should be equivalent

And ignore the second to last line lol

[vocal sleet]

@lofty agate Has your question been resolved?

[lofty agate]

Hmm, ok, thanks

[vocal sleet]

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[hearty flare]

.close

[vocal sleet]

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[deep bough]

2 questions on the text highlighted in red

the radius of V is not specified but we assume it's less than delta correct?

also, for f to be continuous don't we have to say for all epsilon > 0 and not there exist an epsilon > 0
how come we're allowed to "stick" with "there exists and epsilon"?

lastly for the text highlighted in yellow
Am I correct to assume that "as soon as" was used in the past more often and now for current proofs we would use different language, right?
instead of "as soon as" I could replace it with "if"

[gaunt sparrow]

epsilon is the radius of a ball in V.
delta is the radius of a ball in X.

The point of the first part is that given a point p in X, since V is open, we can find an open ball that contains f(p), and then by continuity we can track that to a ball in X.

[gaunt sparrow]

If you mean when they write "Since V is open, there exists epsilon > 0...", then they aren't using continuity here, just the fact that V is open.

[gaunt sparrow]

I wouldn't say that it was used more before. There's tons of other ways of writing this. "if" works, "whenever" also works. It's more stylistic than anything.

[deep bough]

so epsilon is the radius of a ball in V and since V is open then f(p) is an interior point
by definition, f(p) is an interior point if there exist a neighborhood contained in V
d(f(p), y) < epsilon shows that there is a neighborhood of f(p) contained in V
is my line of reasoning correct?
asking because of the line " epsilon is the radius of a ball in V"
this is the way my head is thinking about that line and I want to know if it's correct

[gaunt sparrow]

Yes that's correct. Since V is open, any point in V is interior and so there has to be some open ball centered at that point that fits within V. We pick epsilon to be the radius of that open ball.

[deep bough]

oh and one last question

f is a mapping from X INTO Y
the INTO is why we're allowed to say f(x) in V implies x in f^-1(V), right?

I've been taught to see onto as surjective and into as one to one which is why I think this

[gaunt sparrow]

Hum, I get what you mean, but in this case I think it`s just the way it's written. You don't need continuous maps to be injective/surjective.

If f-1(V) was empty, then there's just nothing to prove, since the empty set is open.

I guess the proof could specify that if f-1(V) is empty, then this vacuously holds, but it's somewhat unnecessary.

[deep bough]

okay
thank you for all your help
🙂

[gaunt sparrow]

No problem!

[deep bough]

.close

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[hearty flare]

a = (1-sinθ)/(2cosθ) 1/a= 2(1+sinθ)/cosθ

[hearty flare]

find sinθ and cosθ in terms of a

[raw crag]

Divide both the equations to eliminate cos theta

[hearty flare]

i think that we can only use 1 equation

at one time

[vocal sleet]

@hearty flare Has your question been resolved?

[hearty flare]

<@&286206848099549185>

[vocal sleet]

@hearty flare Has your question been resolved?

[twin meteor]

BH

[hearty flare]

.close

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[vocal sleet]

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[manic lance]

how do i change my english text and number to bolder ones? like arial text, something with a more visible font

[fossil shoal]

$\text{abc} ~ 123 ~ \textbf{abc} ~ \mathbf{123}$

[twin meteor]

idioticbaka1824

[manic lance]

how do i globbally enable that?

[fossil shoal]

oh. not sure...

https://tex.stackexchange.com/questions/23957/how-to-set-font-to-arial-throughout-the-entire-document
does this help

[manic lance]

yes this is awesome, can you help me change the font for equations to Arial too?

[fossil shoal]

i got this from a google search

i'm sure you can find something, stackexchange has everything

[manic lance]

ive been looking this entire time and before for hours and still couldnt find one, so thats why im asking you

[fossil shoal]

https://tex.stackexchange.com/questions/469775/how-to-use-math-mode-and-siunitx-with-arial
https://tex.stackexchange.com/questions/668939/change-math-font-to-arial

[manic lance]

these dont work for me

[fossil shoal]

https://www.overleaf.com/learn/latex/Font_typefaces

if you're just looking for a font with better visibility, maybe you can pick one from here

[twin meteor]

I don't recognise the option onfig. Use ,texconfig to see the list of options.

Jul 5, 2024, 6:32 AM

Personal LaTeX configuration.

​ ​ ​ ​ ​ ​ autotex: Disabled (may be overriden by guild settings)
keepsourcefor: Don't delete source (may be overriden by the guild)
​ ​ ​ ​ ​ ​ ​ colour: Using the white colourscheme
​ ​ ​ alwaysmath: Disabled
​ ​ ​ alwayswide: Disabled
​ ​ ​ ​ namestyle: DISPLAYNAME
autotex_level: WEAK

To see more detailed information use ,texconfig <option>.
To set an option use ,texconfig <option> <value>.

Preamble

Using a custom personal preamble with 46 lines!
Use ,preamble to view or modify your preamble!

Jul 5, 2024, 6:32 AM

[manic lance]

i saw it already, the point is to set it up as defualt for equations, also these suck

[fossil shoal]

\usepackage{fontspec}
\setmainfont{Cambria}[BoldFont = CambriaB]
\usepackage{unicode-math}
\setmathfont{Cambria Math}

try this

[left portal]

@fossil shoal@fossil shoal

[manic lance]

it crashes

[fossil shoal]

hmm okay i don't think i can help you sorry

[manic lance]

it's alright, ty

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@manic lance Has your question been resolved?

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[light mantle]

hello everyone

[warm grove]

hi

[vocal sleet]

@light mantle Has your question been resolved?

[drifting scroll]

what

[fervent wasp]

.close

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[vast shale]

,tc color dark

[twin meteor]

You have switched to the dark colourscheme.

[vast shale]

Let $B=A^3-2A^2+3A-I$ where $I$ is a unit matrix and $A=\begin{bmatrix}
1 & 3 & 2\
2 & 0 & 3\
1 & -1 & 1
\end{bmatrix}$ then the transpose of matrix $B$ is equal to

[twin meteor]

pun pun

[vast shale]

how can i find B using characteristic equation

[drifting scroll]

,tc color dark

[twin meteor]

You have switched to the dark colourscheme.

[vast shale]

nvm got it

.close

[vocal sleet]

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[vocal sleet]

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[manic lance]

how to show when a_n->+inf this is true?

[oak magnet]

What is an?

[manic lance]

a sequence

[oak magnet]

Any ?

[manic lance]

general a_n that lim a_n = +inf

[tidal dock]

how would you show if it was just n and not a_n

[manic lance]

idk, just say it's true?

[tidal dock]

just trying to understand what tools you are supposed to use

[manic lance]

like listen the previous question was very long to prove and find a lim and all, and this is just the 2nd question now so I think it's more than just a 1-liner

[manic lance]

yeah me too

[oak magnet]

So you have the limit of (1+1/n)^n ?

[manic lance]

I mean I could maybe say lim of the above equals lim of the same expression with a_n=n since they have the same limits and end it there I guess?

[manic lance]

a_n isn't necessarily =n

[tidal dock]

ooo

maybe epsilon delta?

idk

[manic lance]

but this is a sequence not a function

[tidal dock]

like you can find N s.t. with a_N the limit is arbitrarily close to e

[oak magnet]

They are talking about cauchy

[tidal dock]

yeah that's kinda the idea

[manic lance]

I don't understand the terminology, I do wanna use squeeze theorem if I can though

[manic lance]

no one said the given expression is a cuchy sequence, also idk how to prove it is

[oak magnet]

It leads to "Block" a_m between the two expression where their limit is e both

[manic lance]

how do I prove?

[oak magnet]

It will lead to find out its a cauchy one no ?

[tidal dock]

yeah it is a cauchy sequence

[oak magnet]

https://math.stackexchange.com/questions/1450946/how-can-i-prove-lim-limits-n-to-infty-left1-frac1a-n-righta-n-e

[oak magnet]

Yeye

[tidal dock]

and I'm pretty certain $\bN \cup {-\infty, +\infty}$ is compact

[twin meteor]

artemetra

[manic lance]

.close

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[vocal sleet]

Please don't occupy multiple help channels.

[drifting scroll]

what have you tried? we're not a do-your-homework channel

[rustic juniper]

oop

i did all the problems before sending them

i just wanna make sure i understand

i have tried

[vocal sleet]

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[finite trout]

I can't figure out how to do this

[finite trout]

I tried but got it wrong

[stone gazelle]

You were on the right tracks here

[split chasm]

Square both sides

[drifting scroll]

and u will be done

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@finite trout Has your question been resolved?

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[ashen lake]

i need help

[drifting scroll]

!status

[vocal sleet]

What step are you on?
1. I don't know where to begin.
2. I have begun but got stuck midway.
3. I got an answer but I was told that it's wrong.
4. I got an answer and would like my work checked.
5. I have a question about someone else's work/solution.
6. I have completed the problem and don't need help anymore. Thank you.
7. None of the above

[ashen lake]

1

[drifting scroll]

and!?

[ashen lake]

misstype

sorreh

[ashen lake]

a grid

hmm so does 51 points represent the whole square or only 1/25

[drifting scroll]

if you can get each circle to cover a square, then you can use pigeonhole

[ashen lake]

ok i confuse this part

so imma assume 51 points is pigeonholes and n = pigeons

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@ashen lake Has your question been resolved?

[ashen lake]

<@&286206848099549185>

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[ruby zinc]

hello i need help with an algebra problem

[ruby zinc]

i need to find the possible values of this

<@&286206848099549185>

.close

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[atomic pasture]

<@&286206848099549185>

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[terse kite]

how can i achieve a function/rearrange this equation to discover x?

[terse kite]

i tried searching for exponent properties that i can use to rearrange this but couldn't find any useful ones.

[placid spear]

logarithm?

you can try convert it to: $2 = 8^{\frac{1}{x}}$

[twin meteor]

Closer

[terse kite]

ty 😃 👍

i knew that existed, didnt know what it does tho

[placid spear]

And then using logarithm

[terse kite]

alr

.close

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[lost ravine]

find the roots of the equation $3-|x|=|x-1|$

[twin meteor]

monul

[lost ravine]

so. the correct answer is (2:-1) . I have a solution, but I'm not sure if it's the right one.

[vocal sleet]

@lost ravine Has your question been resolved?

[loud walrus]

Show your work

[lost ravine]

2. No root

[sudden compass]

just plot a graph

replace 3 with y

and then draw the graph of y=3

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[vocal sleet]

To ask for mathematics help on this server, please open your own help channel or help thread. See #❓how-to-get-help for instructions.

[vast shale]

post the question bruh

[tranquil trellis]

do you know the conversion from cups to teaspoons

[vast shale]

wth is this sort of question

[weary inlet]

i think it’s just ratios?

[vast shale]

we're never taught these conversion...

[weary inlet]

like x/2 = 17.5/7

it’s worded badly but i think that’s what they want?

idk

[vast shale]

hello teachers....

I wanted a help

we're taught integration, differentiation, by our physics teacher... but don't have these topic this year and I wanted to have a good grip in these topics

[weary inlet]

open another help channel

[vast shale]

why😭

[hushed pewter]

!occupied

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Someone else is already using this help channel. If you need help with a question, please open your own help channel/thread (see #❓how-to-get-help for instructions).

[vast shale]

I didn't know about it, sorry

[hushed pewter]

.close

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[dim hollow]

Hello,

Can someone please help me prove the circle property that the perpendicular bisector of any chord intersects the center of the circle?

[dim hollow]

(i am bored in summer math school, don't worry, i did not fail)

[tidal dock]

!status

[vocal sleet]

What step are you on?
1. I don't know where to begin.
2. I have begun but got stuck midway.
3. I got an answer but I was told that it's wrong.
4. I got an answer and would like my work checked.
5. I have a question about someone else's work/solution.
6. I have completed the problem and don't need help anymore. Thank you.
7. None of the above

[tranquil trellis]

Doesn't this follow from circumcenter properties

[dim hollow]

i tried making similar triangles paralell to the diameter

[tidal dock]

show your progress, then

[dim hollow]

the problem is, i cant prove that the intersection point from the chord is equal on both sides of the smaller triangle

[tranquil trellis]

show your work so we know what triagnles you're talking about

[dim hollow]

not convenient rn

[tidal dock]

huh? doesn't that follow from it being a bisector

[dim hollow]

i can't prove that the stuff in red is equal to each other

[tranquil trellis]

Do you know the property that the perpendicular bisector is the set of points equidistant from the two points that make the line segment

[dim hollow]

yes

[tranquil trellis]

And is the center equidistant from the two endpoints of the chord

[vocal sleet]

@dim hollow Has your question been resolved?

[vocal sleet]

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[deep bough]

Wanted to know if the proof for number 7 is correct

c is the set of all Cauchy sequences and l^(infinity) is the set of all bounded sequences

<@&286206848099549185>

[ancient adder]

hopefully i wont study these in my univ

hey bro im gona study CS

will i study these?

univ

[deep bough]

Most likely not

[vast shale]

isn’t that mostly economics

That uses it

[vast shale]

ur gonna use a lot of sequences and series

[ancient adder]

im fked up

i skipped those

[vocal sleet]

@deep bough Has your question been resolved?

[deep bough]

it is what it is... 😦

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@deep bough Has your question been resolved?

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[wary mantle]

#linear-algebra message

[blissful sentinel]

(Please don't cross-post the same q in multiple places)

What's wrong with just looking up the definition though as Denascite said lol

Oh, looks like your q has been answered anyways

.close

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[thin vale]

I'm trying to compute the variance

I thought that E[X^2] = 1 * 1/2 + 4 * 1/2^4 + 9* 1/2^9 + ....

So I get this sum for that term which I know converges via ratio test but I don't know how to compute its value

[urban edge]

i would assume you dont have to open more than four boxes here

why is it infinite

[thin vale]

you open them randomly

it doesnt say once one is open you remove it

[urban edge]

oh weird

[thin vale]

I assume we just close it back up and possibly reopen it

[urban edge]

also why does it say price in line 3

[thin vale]

I assume it means prize lol

[urban edge]

what if you dont close the box

[thin vale]

well then its finite and trivial

[urban edge]

it seems overly complicated to do that infinite sum

[bitter pilot]

yea

[urban edge]

,w sum 1 to inf of (n^2)/(2^(n^2))

[bitter pilot]

suggesting me to compute it

[twin meteor]

Results provided by WolframAlpha

Click here to refine your query online
Upgrade to WolframAlpha Pro!

Requested by garlicbredfries

[urban edge]

wolfy can only approximate

[bitter pilot]

post the original bro

[urban edge]

the original is posted

[bitter pilot]

it just appeared thx

[urban edge]

tbf it doesnt say you close the boxes so im gonna believe you cant open boxes infinitely

[thin vale]

you underestimate my dedication to finding this prize

[bitter pilot]

closing the boxes is kind of ehh

you still know that this box has no price after opening it once

so why would you open it infinite times

[thin vale]

Because the problem says you pick randomly

[urban edge]

and then open the box

you do not close the box

[bitter pilot]

yea but sooner or later if you opened at least 3 boxes then in the 4th there will be a price

infinite times doesnt make sense

[urban edge]

i love you dedication to calling them prices

isnt this your intro probability/stats class anyway ossie

if anything, the ultimate source is asking your prof

[thin vale]

it does if u close the boxes and pick randomly again

😠

[thin vale]

just probability

[sly sierra]

why the hell would you close the box and pick randomly again

[bitter pilot]

but do you change the order of boxes everytime you open

[sly sierra]

it makes no sense to interpret it that way

[thin vale]

it says

"randomly"

[bitter pilot]

do you throw them into the air

[thin vale]

if I throw out boxes i've already opened

[sly sierra]

sure, randomly from among the remaining boxes

[thin vale]

its a strategy

[thin vale]

it doesnt say from the remaining

😠

[bitter pilot]

i believe you are making it a bigger thing than it is

[thin vale]

😠

[bitter pilot]

like your bench press

[sly sierra]

well feel free to interpret it the insane way if you want, it's your grade haha

[thin vale]

my bench press is huge

[bitter pilot]

(joke)

[thin vale]

okay now im just curious about how one would compute this series

[urban edge]

seen as wolfy approximates, its not computable

it does converges quickly with O(1/(2^(n^2)))

[thin vale]

https://tenor.com/view/dog-doggie-trained-dog-dog-training-dog-carry-gif-12125736216390932776

Okay

.clos

.close

[vocal sleet]

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[bitter pilot]

I would interpret it that way if it was a 2 hour exam and thats the only question

[sly sierra]

fortunately in that context i guess you could just ask

in this context too, for that matter

[thin vale]

dont worry sirs

I have asked

[livid tapir]

huh why is it 1/2^(n^2)

wouldn't is just be 1/2^n

[thin vale]

X^2 takes on values 1, 4, 9, 16, ...

so its expectation is 1 * P(1) + 4 * P(4) + ..

and P(n)= 1/2^n

[livid tapir]

P(X^2 = n^2)

hmm not the right explanation but

This is the definition

[sly sierra]

yea denom is wrong

[thin vale]

hm okay well my sir wrote back already

[sly sierra]

your sir speaks wisely

[thin vale]

why is this?

[rugged orchid]

That’s a definition

[livid tapir]

I think it's just the definition of E[g(X)]

[rugged orchid]

Wdym why

[rugged orchid]

Tis called lotus