help-36

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[frank cloud]

Prove by contradiction, that there is no least positive rational number

[frank cloud]

I sort of know how to do it, but what would be the proper notation for that

I had the idea that any positive rational number divided by an integer greater than 1 (e.g. 2) would produce a smaller number so we could go on forever

So the only thing im having tousle with is the notation

[hybrid heath]

sounds good

You basically have it exactly like that

Assume there is some least positive rational number p

show p/2 is rational and 0<p/2<p and that's your contradiction

[frank cloud]

Oh

Sounds straight forwad

Thanks

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[lunar forum]

To be honest, I don't know where to start with this problem

If anyone can give me a hint

Wouldn't this be the union of rational numbers?

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[fallen comet]

idk where to start

[fallen comet]

<@&286206848099549185>

[torn rock]

ibox me

inbox for full answer

[ivory vessel]

!nosols

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As a helper, please do not give out answers that could be copied as a homework solution. Have the student work through the problem themselves and guide them along the way.

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[woven forum]

i got a+b=3 but dont know what to do after that

[ivory vessel]

they are touching, meaning their tangent at x=1 must also have the same gradient

in other words, you also need to calculate f'(1) = g'(1)

[torpid sequoia]

you also have to constrain f(1) = g(1) = 3

[woven forum]

ohhhh

got it thanks

[ivory vessel]

yup, thats how he got a + b = 3

[torpid sequoia]

oh i missed that sorry

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[little flame]

What do you think about task number 11?

[little flame]

I personally think that I have no idea how to solve it, so suggestions?

[tranquil pine]

use S_n - S_n-1 = a_n

[little flame]

Oh thank you I will give it a try

[tranquil pine]

not corret

?

(Sn - Sn-1) /n

that is correct An

its not the same a_n

Look at {An}

there is k *Ak

so need /n

[tranquil pine]

.

please read the messages

and i suggest deleting your messages as well

[little flame]

I got a_20=a_n

[tranquil pine]

oh no

[little flame]

smth wrong

[tranquil pine]

Sn - Sn-1?

[little flame]

Yes

[tranquil pine]

Sn - Sn-1 = ?

[little flame]

a_n

[tranquil pine]

Sn = n^2 + 2n

right?

[little flame]

yes

[tranquil pine]

Sn-1 = (n-1)^2 + 2(n-1) ?

[little flame]

one sec

[little flame]

How did you get that

[tranquil pine]

oh no.....

how can u get Sn?

[little flame]

n(n+1)/2

@tranquil pine

[pastel cosmos]

what

no

we're letting S_n = a_1 + 2a_2 + ... + na_n

so you can't use that formula

[pastel cosmos]

right?

[tranquil pine]

yes

[tranquil pine]

oh no

[pastel cosmos]

you're using wrong formula

[little flame]

I see

[pastel cosmos]

That formula means 1 + 2 + 3 + ... + n = n(n+1)/2

But we're letting S_n = a_1 + 2a_2 + ... + na_n

[little flame]

so?

[pastel cosmos]

which is S_n = left hand side

[little flame]

What will be S_n-1?

[pastel cosmos]

for example

S_ 1 = a_1

S_3 = a_1 + 2a_2 + 3a_3

right?

[little flame]

yes

Oh I get it

S_19

20a_20=a_n?

[pastel cosmos]

So S_(n-1) = a_1 + 2a_2 + ... + (n-1)a_(n-1)

Then what is S_n - S_(n-1)

[little flame]

If we subtract S_20-S_19=a_n?

[pastel cosmos]

no

Let's see again

S_20 = a_1 + 2a_2 + 3a_3 + ... + 19a_19 + 20a_20

S_19 = a_1 + 2a_2 + 3a_3 + ... + 19a_19

[little flame]

20a_20

[pastel cosmos]

yes

there we go

but on the other way

S_20 = 20^2 + 2*20

you can compute that

and S_19 = 19^2 + 2*19

[little flame]

S_20-S_19=20a_20

[pastel cosmos]

yes

[pastel cosmos]

because of the given equality

[little flame]

Ah I get it now

[pastel cosmos]

yay

[little flame]

Thank you

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[pearl spear]

can someone help me oslve this

[junior radish]

Simplify x, then replace in x² - 2ax

[pearl spear]

welp i did

i squared both sides

[junior radish]

Can you show?

[pearl spear]

uhm, i dont have discord on mobile

basically

[junior radish]

That's not really helpful I think

[pearl spear]

i squared both sides

[junior radish]

Rationalize is a better option

[pearl spear]

ooo

i didnt think abt that

[ornate knot]

it would be better if u multiply and divide with conjugate

[pearl spear]

alr

ima try

yes its solved

ty

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[clear crown]

can anyone see the solution here?

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@clear crown Has your question been resolved?

[tawdry storm]

Is this supposed to be a pattern recognition problem, or do you have a formula somewhere?

Because I'm not seeing a pattern

[clear crown]

no formula

got to find a patern myself

but i can't see either

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[unreal notch]

<@&286206848099549185>

[granite gazelle]

are there any high school problems here

like bro im in eighth grade 1tf is happening

[clear crown]

lol

this isn't exactly a math problem

[clear crown]

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[normal hill]

Ok so big question

[normal hill]

we have the ODE y'' + 2y' + y = 0

I want to find power sereis solutinos

we try

$y(x) = \sum_{n=0}a_nx^n$

[soft zealot]

Mortta

[normal hill]

As our trial

Doing that i got

https://gyazo.com/ec4e2498faf64b9b4b59afd3aebbbf3c

We than just got the answer using general solution of ODE

and we got

y = C1e^-x + xC2e^-x

So how tf do i do this:
https://gyazo.com/4337ad33e184ce7bbea65ece3d89b1f6

idk what to do

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[latent bane]

im trying to proof this but i get stuck with just cosines, using cubic sum, what else can i try

[soft zealot]

Roman_Garland

Roman_Garland
Compile Error! Click the reaction for more information.
(You may edit your message to recompile.)

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[earnest shale]

Hi can I get help with c

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[warm python]

$\int \frac{\arctan\left(x\right)}{\left(1+x\right)^3}dx$

[soft zealot]

Why am. I here

[warm python]

can I have a hint

$x=tan(\theta)$ doesn't work vey well

[soft zealot]

Why am. I here

[warm python]

*very

[rustic musk]

I have no idea but an integration by parts might make sense

[warm python]

Hmm, I'lll try that

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[warm python]

Thanks

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[scenic rapids]

does any1 happen to have a counter example for this...

[scenic rapids]

i know its false but cant seem to find a ceg ughhhh

[smoky plover]

$f(x) = 6x^3 \ f(x) = 3x\times 2x^2 \ f(x) = 6x\times x^2$

[soft zealot]

Adam Chebil

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[scenic rapids]

this is wht happens when u overthink.... thank you so much ;-;

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[hidden kiln]

Can someone help me with this question pls

[hidden kiln]

But I don’t want the actual answer, I just need some help

I don’t know where to start

[tribal jasper]

idk if it's right but i might be able to get you pointed in the right direction

since p, q, r, s, t are consecutive positive integrs

using another integer n (n≧3)

we can write it as
p=n-2
q=n-1
r=n
s=n+1
t=n+2

p+q+r+s+t=5n
q+r+s=3n

all we have to do now is find an integer n so that 5n is a perfect cube and 3n is a perfect square

I did find one that works, just not too sure if it's the smallest

[hidden kiln]

Why can’t we set p=n, q=n+1, r=n+2 and so on

[tribal jasper]

your idea is totally valid

but if we do n, n+1, n+2, n+3, m+4

when we add p+q+r+s+t it becomes 5n+10

q+r+s becomes 3n+6

and it makes it harder to check if it's a perfect cube/square

because of the +3 and the +5 at the end

[hidden kiln]

Wouldn’t it be 5n+10

[tribal jasper]

oh yeah

youre right its 5n+10

sorry bout that

and its 3n+6

[hidden kiln]

But what’s wrong with that

[tribal jasper]

We're being asked for the value of R

and it's wiser to set n as R so we don't have to do any tinkering later

[hidden kiln]

Oh ye sry

[tribal jasper]

also doing it like I did makes it so both p+q+r+s+t and q+r+s are in* multiplication form only*

sorry if what I'm saying is not making sense

[hidden kiln]

And as r is a positive integer n can’t be negative meaning that n can’t be any smaller negative number

[tribal jasper]

yes exactly

hence the n≧3

[hidden kiln]

But p+q+r+s+t is a perfect cube

So I guess that we can’t assume that p is 1

[tribal jasper]

yeah

if p=1 then 1+2+3+4+5=15 which isn't a perfect cube

[hidden kiln]

So we need to find what is the least value that p can be

[tribal jasper]

i think we're supposed to find the smallest r

from the screenshot you posted

[hidden kiln]

But for that we need p to be smaller but also pqrst added together should give u a perfect cube

So p<q<r<s<t

As they r consecutive

[tribal jasper]

I did it as
p=n-2
q=n-1
r=n
s=n+1
t=n+2

and if we find the smallest n

p should be the smallest as well

since p is 2 less than r

[hidden kiln]

Ye

So p<r

[tribal jasper]

yes

[hidden kiln]

So what’s next

[tribal jasper]

Using this new way to write p, q, r, s, t

If you add p, q, r, s, t

p+q+r+s+t=
n-2
+n-1
+n
+n+1
+n+2

which equals to 5n

and q+r+s=
n-1
+n
+n+1

which equals to 3n

[hidden kiln]

Ye

[tribal jasper]

Now all you have to do

is find the smallest n

so that 5n becomes a perfect cube

and 3n becomes a perfect square

[hidden kiln]

Ye

So would it be a trial and error method

[tribal jasper]

Yeah

The last part I couldn't figure out logically

or

[hidden kiln]

K lemme try

[tribal jasper]

oke 👍

sorry for the long explanation

[hidden kiln]

It’s ok

[tribal jasper]

❤️

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[hidden kiln]

.reopen

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✅

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@hidden kiln Has your question been resolved?

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@hidden kiln Has your question been resolved?

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@hidden kiln Has your question been resolved?

[hidden kiln]

@tribal jasper have a look at this similar type of question

answer:

can we not do it this way?

<@&286206848099549185>

[ionic venture]

Where do you need help?

[hidden kiln]

^^^

<@&286206848099549185>

[final saddle]

@hidden kiln Has your question been resolved?

[digital steeple]

whats your question?

[hidden kiln]

^^^

[digital steeple]

i have seen that. and i have seen that you posted also the solution. so i ask you again, whats your question?

[hidden kiln]

What r u talking about?

i don't know the solution

[digital steeple]

wtf? what is this?

[hidden kiln]

and i never posted any solution related to this quesiton^^^

[hidden kiln]

this is the solution for a similar type of question

[digital steeple]

explain the difference.

[bleak arch]

Yeah you solve in terms of the value you are finding

They are consecutive integers

[hidden kiln]

?

[digital steeple]

you say its a similar example. i cant see a dfiference.

[hidden kiln]

i know that this question looks very similar, but this quesiton did not say that the consecutive positive integers are in ascending order

[hidden kiln]

^^^

This is the actual question that i need help with

[digital steeple]

you have 5 conesecutive positive numbers in increasing order.

The solution you posted starts with:

in my world does this mean p,q,r,s,t are increasing. exactly what you need.

[hidden kiln]

but i also have some doubts in the solution that i posted

[digital steeple]

so again: what is your question? what are your doubts in the solution?

[hidden kiln]

i don't understand this part

[digital steeple]

q+r+s = 3r - right?

[hidden kiln]

like how is it 3^2n+1

and 5^3m+2

[hidden kiln]

no i get this part

[digital steeple]

q+r+s = 3r, this is a perfect square and its divisible by 3. so you need an even count of the prime factor 3. one comes from 3, and - to be even - and odd count must come from r. -> 3^(2n+1) divides r.
and the same with 5.

[hidden kiln]

i get how it's divisible by 3

but not the rest

[digital steeple]

what exactly?

[hidden kiln]

"so you need an even count of the prime factor 3. one comes from 3, and - to be even - and odd count must come from r. -> 3^(2n+1) divides r."

[digital steeple]

is 3 times 7 a perfect square?

[hidden kiln]

no

[digital steeple]

why not?

[hidden kiln]

cuz 3*7=21 and sqr root of 21 is not an integer

[digital steeple]

what do you have to change/to add that it becomes a perfect square?

[hidden kiln]

what becomes perfect sqr

[digital steeple]

3 times (7x) should be a perfect square.

what has x to be?

in a minimal version.

[hidden kiln]

idk

21x should be a perfect square

?

[digital steeple]

if a perfect sqare is divisible by a primefactor, it has to be divisible by primefactor^2. so here it si divisible by 3, so it must be divisible by 9.

(3r) is divisible by 3 and it is a perfect square, so r must be divisible by 3.

[hidden kiln]

ye

[digital steeple]

what if r is divisible by 9? is that possible?

[hidden kiln]

i don't think so

[digital steeple]

if r is divisible by 9 -> (3r) is divisible by 27, as (3r) is a perfect square, it must be divisible by 81. -> r must by divisible by 27. so if r is divisible by 9, r has to be divisible by 27.

[hidden kiln]

oh ye 9(3r)=27r

[digital steeple]

so, you know if 3^(2n) divides r, 3^(2n+1) must divde 3.

[digital steeple]

and at the end its still this argument.

[digital steeple]

and this is still the solution for your example.

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[tranquil pine]

Okay, I evaluated R'(3)

not exactly sure how to interpret it, or put it into words exactly

[ashen warren]

whats the ' mean

[tranquil pine]

R'(3) = 1150, so i said "that's the rate of revenue change when increasing the price 3 times"

[tranquil pine]

Prime. Derivative of the original function

[ashen warren]

i have no clue what that menas

[tranquil pine]

then why dafuq are u here goofy

[vapid basalt]

It's basically the rate your revenue will change based on how many customers you lose

Sorry based on the price increase

But it works the same either way

[tranquil pine]

Alr so my explanation is correct

[vapid basalt]

Nope, it's how much your revenue will change if you increase your price by 7.50

X is the amount of 2.50$ increments

[tranquil pine]

Ahh so R'(3) tells us that the revenue will increase by $1150 with 3 price increases?

[vapid basalt]

Yeah

[tranquil pine]

alr

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[true walrus]

I don't need hlep figuring out the equation just understanding math lingo, what does the R -> R thing mean?

[true walrus]

I know that fancy R is ment to represent real numbers but what does the arrow mean?

[crude wagon]

@true walrus

[true walrus]

yes?

[dapper bramble]

it means f is a function with domain and codomain R

[true walrus]

domain is the main thing

codomain is the second thing right

[dapper bramble]

yes

[true walrus]

so the domain is R and the codomain is R

doesn't that just mean all real numbers

[dapper bramble]

yea R is the set of real numbers

[crude wagon]

Range is set of outputs under the function

may not always be same as codomain

[true walrus]

yea range is a whole different thing now

wild.

[dapper bramble]

range is sometimes used to mean codomain

[crude wagon]

Range is a subset of co-domain

[dapper bramble]

which i don’t like bc it’s unclear

but people do it

[true walrus]

mathematicians have no imagination fr

can't think of a simpler way to show this

so anyways

if it showed it like R -> Z, does that mean Z is the codomain of R?

[dapper bramble]

codomain of the function you would say

[true walrus]

the function f(x)

then whats the point of R -> Z??

[dapper bramble]

the function f

[true walrus]

this uni stuff is harder than I thought it would be to learn

[dapper bramble]

ok like yea i understand how it could seem kinda pointless to have a codomain at all

but it matters when you want to for example compose functions together

[true walrus]

do you mind providing an example of what you mean?

[dapper bramble]

like to describe when it makes sense to compose functions

[true walrus]

ohh

so let me just straighten the definition out for me to understand:
Domain is all the numbers that can go into the function
the codomain is all the numbers that can come out of the function
the range is the numbers that the domain has taken?

[dapper bramble]

no

well

ok the last thing is weirdly worded but maybe

[true walrus]

I am not sure how to explain it but

the codomain is like every single outcome from the domain but the range only takes a few of them from the codomain

[dapper bramble]

the codomain is just something you define when you define the function

[true walrus]

hmmm

so the codomain is the possible outcomes of the function

[dapper bramble]

depends what possible outcomes means

the range is the set of outputs you actually get from passing elements through the function

[true walrus]

then the codomain is the set of possible outputs of passing hypothetical elements through a function?

[dapper bramble]

umm

no that just sounds like the range

[true walrus]

ok nvm

so to loop back to the original

how to I say R -> Z

like how we call > 'greater than' do u understand?

[dapper bramble]

that doesn't make much sense by itself

f: R -> Z means f is a function with domain R and codomain Z

[true walrus]

ahh

ok

[dapper bramble]

when you do this

[true walrus]

that makes sense now

[dapper bramble]

you can then ask questions like, is the range (or image) of the function equal to its codomain?

[true walrus]

I see

[dapper bramble]

in other words do you hit everything in the codomain when you pass elements through f

[true walrus]

I understand now, thank you for the help

[dapper bramble]

yw

[true walrus]

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[uneven temple]

how do i make my casio calculator give answers in surd form

[uneven temple]

my friends have similar calculators and have their answers in surd form

idk ifmy calculators just old but its literally the same calculator

surely someone knows

[thorny tendon]

that'd prolly be a question for the company who made your calculator

[uneven temple]

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[uneven temple]

/close

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[final saddle]

Please don't occupy multiple help channels.

@lone elm Has your question been resolved?

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[vocal nimbus]

Numbers (a), (b), (c), (\frac{ab}{c} + \frac{ac}{b} + \frac{bc}{a}) are integers. Are all three numbers (\frac{ab}{c}), (\frac{ac}{b}), (\frac{bc}{a}) necessarily integers?

[soft zealot]

kitten

[minor mist]

yes

integer + integer = integer

oh wait

[vocal nimbus]

It's only given that sum is integer

[minor mist]

nah

take for example

1/3

its not

true

[midnight sparrow]

? why explain

[minor mist]

ab/c = 1/3

nvm

im trippin

hold on

[vocal nimbus]

<@&286206848099549185>

[cerulean radish]

woah

[inland kettle]

!15m

[final saddle]

Please only use the <@&286206848099549185> ping once if your question has not been answered for 15 minutes. Please do not ping or DM individual users about your question.

[cerulean radish]

discord kitten

:0

WHAT

[midnight sparrow]

I think not necessarily think of roots of a polynomial

a polynomial of degree 3 with coefficients integers here doesn't mean all his roots are integers

[vocal nimbus]

Also last number is integer

[midnight sparrow]

rational root theorem can help you

[vocal nimbus]

$\frac{ab}{c} + \frac{ac}{b} + \frac{bc}{a}$

[cerulean radish]

u missed the

[soft zealot]

kitten

[cerulean radish]

dollars

[minor mist]

therefore it has to be an integer?

[midnight sparrow]

?

[minor mist]

product is an integer

[midnight sparrow]

and ?

[minor mist]

sum is also an integer

[vocal nimbus]

It doesn't imply that

[minor mist]

contradict?

[midnight sparrow]

$1/3*6=2$

[minor mist]

example

[soft zealot]

phoestaclies

[minor mist]

okay

[minor mist]

but the sum?

the sum isnt an integer

[vocal nimbus]

<@&286206848099549185>

[minor mist]

the sum as well as product is an integer

[cerulean radish]

!15m

[final saddle]

Please only use the <@&286206848099549185> ping once if your question has not been answered for 15 minutes. Please do not ping or DM individual users about your question.

[cerulean radish]

jesus

u pinged us once

dayum

[vocal nimbus]

It was already 15 mins

[midnight sparrow]

calm I'll help you

[minor mist]

@midnight sparrow

if the sum and product of the three numbers is an integer then I believe those 3 numbers must be an integer?(wrt question)

[ionic venture]

Let ab = x, bc = y, ac = z

[vocal nimbus]

Yea and that's why we can't say that it's integer if product is integer...

[minor mist]

why cant you?

product is abc

a,b and c are integers

[inland kettle]

STOP PINGING US

I BEG

[vocal nimbus]

The task needs another proof

[ionic venture]

$Hello$

Hello

[vocal nimbus]

You don't even help

[soft zealot]

Roman_Garland

[midnight sparrow]

$\sqrt 2 - \sqrt 2 + 0 = 0$ and the product is 0

[soft zealot]

phoestaclies

Roman_Garland

[minor mist]

root2 isnt integer

[midnight sparrow]

you didn't precise this in your question

[minor mist]

and 0 isnt valid

[midnight sparrow]

you want to prove that it's integer so affirm that it's integer is contradictory

[minor mist]

....

the question's here

[soft zealot]

Roman_Garland

[vocal nimbus]

4 - sqrt(3), 4+sqrt(3), 1

[minor mist]

dude

a,b and c are INTEGERS

[vocal nimbus]

SO?

[minor mist]

root3 is an integer?

[vocal nimbus]

it's not related to question

[minor mist]

im talking wrt

question

my

bad

i didnt specify

[vocal nimbus]

we are given that a, b, c and $\frac{ab}{c} + \frac{ac}{b} + \frac{bc}{a}$ are integers

[soft zealot]

kitten

[minor mist]

yes

[vocal nimbus]

it's different

but we aren

't given that like ab/c is integer, we need this to prove

[minor mist]

oh yeah prove

logically integer

[vocal nimbus]

Yea but proof is required

Oh, I see, thank you

.close

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[midnight current]

Hello how do u do (ii)?

[tranquil pine]

,rccw

[tranquil pine]

remember that [
\m\sin{a-b} =\m\sin a \m\cos b -\m\cos a\m\sin b
]

[midnight current]

oh ya then how do u get it to tanx?

[tranquil pine]

well like

what does tanx represent in terms of the sides of a right angle, if you remember?

[midnight current]

like opp/adj?

[tranquil pine]

yeah!

so tanx = opp/adj

now like use Pythagorean theorem to find the hypotenuse

you know the opposite and adjacent

[midnight current]

ohhh okok thanks

.close

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[normal hill]

My prof

is wrong

i subbed in r0 into the top

you get positive epislon for binding energy

not negative

[lucid tree]

what is this? the lennard jones potential?

whats ur question

[normal hill]

Im asking

is my prof wrong

I think he is

and yes it is

[livid brook]

he's asking what is this?

[lucid tree]

he is wrrong

[normal hill]

should be positive correct

[lucid tree]

binding energy is a neg quantity

[normal hill]

coz the minus cancel eachother

[lucid tree]

yeah

[normal hill]

Yea okay thanks

.close

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[sonic crystal]

19. Find the volume of the solid generated when the region under the curve $y=\frac{1}{\left(x^{2}-1\right)}\ \left{\ 2\le x\le3\right}$ is rotated about the line $x=-2$

[soft zealot]

water beam

[sonic crystal]

@lucid tree hello helper

what steps should i do to start

[lucid tree]

hmm do u have the integral expression

i dont remember volumes of revolution that r not at the x or y axis 😭

[sonic crystal]

i thought you were well versed in this area

[lucid tree]

last time i touched calc was 2 years ago

[sonic crystal]

what are you doing now?

[lucid tree]

lin alg

[sonic crystal]

oo

you don’t do any sort of calc in LA?

[lucid tree]

calc is not required in LA at all

u can do a LA course right after learning algebra 2

[sonic crystal]

Oh that’s cool

[sonic crystal]

I’m deciding on what method i should use

hm

cool shape

i think

maybe the radius is 3-x

$V=2\pi\int_{2}^{3}\left(3-x\right)\left(\frac{1}{x^{2}-1}\right)dx$

[soft zealot]

water beam

[sonic crystal]

im thinkin its this

[lucid tree]

ight

now to integrate it

its trivial

[sonic crystal]

what does this mean in a math context

[lucid tree]

u integrate using

partial fraction decomp

[sonic crystal]

but x^2 - 1 cant be factored right

[lucid tree]

it can

(x-1)(x+1)

[sonic crystal]

o

i mustv been thinking of like the other one then

[lucid tree]

x^2+1 can be factored using complex numberds

[sonic crystal]

too scary

[lucid tree]

yeah

[final saddle]

@sonic crystal Has your question been resolved?

[lyric summit]

it is enough to translate by the vector [2,0], i.e. a parallel shift to rotate the graph around the line x = 0, i.e. the Oy axis, etc

[final saddle]

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[mystic zenith]

hey i might be stupid, but im on question 3. like i know the area of a triangle A=l * h/2. but how am i to substitute it with e??

[final saddle]

Please don't occupy multiple help channels.

[pseudo knot]

where's e?

[mystic zenith]

idk, i dont understand it from the question

[pseudo knot]

that's a stylish l to denote each side length

[mystic zenith]

ya. but how does that help me when using area formula

[pseudo knot]

area of equilateral triangle is √3a^2/4

[mystic zenith]

?

[fallen vigil]

where did e come from

[mystic zenith]

how so i though it was base x height/2

[pseudo knot]

you can get to that result by drawing a height line in a triangle

and then using pythagores formula

[mystic zenith]

Oh i use pythagoras

[fallen vigil]

yes

[fallen vigil]

yes

[mystic zenith]

wait how??

where does 3 come from?

like this?

.close

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[fallen rune]

help with this combinatorics question?
I'm using google translate so might have some issues with the translation, on 1 I got $\binom{16}{8}$, and on 2 I got $16\cdot15\cdot \binom{14}{8}$ but I have no idea how to even start tackle 3, it seems really hard

[soft zealot]

horizon2.0

[fallen rune]

just to clarify, the unit includes 10 cells, two of those, the "special cells" the order does matter, while in the other 8 the order does not matter, and the hacker is said to have different "special cells", and that 4 out of the 8 normal cells are exactly like the engineer 8 cells

[final saddle]

@fallen rune Has your question been resolved?

[final saddle]

@fallen rune Has your question been resolved?

[final saddle]

@fallen rune Has your question been resolved?

[final saddle]

@fallen rune Has your question been resolved?

[final saddle]

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[prime cosmos]

can I have help with this?

[lucid tree]

quadratic sequences are of the form
an^2+bn+c

[prime cosmos]

im just stuck on one section

[lucid tree]

ur first step is to take the difference of the sequence twice

[lucid tree]

what section?

[prime cosmos]

i found the difference between 10, 21, 38, 61

[lucid tree]

and 90 too

what is ur new resulting sequence

[prime cosmos]

so the constant difference is +6

and do I need to halve that

so its 3n^2

then im stuck after that

[lucid tree]

okay so u have the coefficient of a, good

now u get 3n^2+bn+c = f(n)

but notice how u have 5 data points?

[prime cosmos]

whats bn + c

and whats f(n)

[lucid tree]

do u know what a quadratic is?

its like

ax^2+bx+c

there is a quadratic term, a linear term, and a constant term

[prime cosmos]

thats the formula

[lucid tree]

yes

so a generic quadratic sequence would have the formula an^2+bn+c

right?

[prime cosmos]

ok

are u substituting n for x right?

[lucid tree]

yeah n and x, it doesnt matter

but what matters is

we know the value of a

u jus found it

a=3

[prime cosmos]

ok

[lucid tree]

so we get 3n^2+bn+c=our sequence

[prime cosmos]

3n^2 +bn +c

yes

[lucid tree]

now when n=1

what is the sequence value?

[prime cosmos]

9

3^2

[lucid tree]

nooo

from the given data

[prime cosmos]

ohh

[lucid tree]

here

[prime cosmos]

so

[lucid tree]

whats the first term

[prime cosmos]

10?

[lucid tree]

yups!

[prime cosmos]

so 3(10)^2

[lucid tree]

so n=1
3(1^2)+b(1)+c=10

[prime cosmos]

oh

[lucid tree]

n is the term u take

[lucid tree]

so this simplifies to?

[prime cosmos]

i was watching this video

and it didn't have this step

lemme show u

[lucid tree]

its what we call systems of equations

[prime cosmos]

https://www.youtube.com/watch?v=xtvJwaYfXss

u can skip to 1:55

u see this part on the video

is whats confusing me

[lucid tree]

they r doing it differently

but its a correct method too

but i think u should consider my algebraic method instead

[prime cosmos]

isn't that way more easy to understand kina

ok

its just kinda hard to understand over message

its fine tho

[lucid tree]

okay so what does this simplify to?

[prime cosmos]

3+ b + c = 10

[lucid tree]

why 103?

[prime cosmos]

accident

[lucid tree]

oh

so thats ur first equation

now lets take n=2

try solving the sequence for n=2 similarly

[prime cosmos]

so for the first equation

3(1)^2

then the next is

3(2)^2

=12

[lucid tree]

nooo

[lucid tree]

this is ur first equation

have u done functions in ur class yet?

[prime cosmos]

no

thats why im doing it

nw

now

ive got a test tomorrow

lmao

[lucid tree]

okay so we know that 3+b+c=10 right based on the given data

what did we do to get this equation?

we let n=1

now at n=2, the sequence is equal to 21

[prime cosmos]

yes

[lucid tree]

so 3(2)^2+b(2)+c=21

what does that simplify to?

[prime cosmos]

ok wait a sec

the part thats confusing me is 3 + b + c = 10

like the b's and c's

[lucid tree]

have u done quadratic equations before?

[prime cosmos]

no

that's what im saying

[lucid tree]

💀

[prime cosmos]

idk it that well

brudda

[lucid tree]

you know how u have linear equations right

y=ax+b

[prime cosmos]

quadratic equations using the formula x = -b rootb^2-4ac/2a

[lucid tree]

thats the solution to the roots

[lucid tree]

but what about the general form?

[prime cosmos]

like is this quadratic equations

[lucid tree]

how does a quadratic function look like?

[lucid tree]

yasss

now u know the value of a by ur difference method

[prime cosmos]

so i substitue

[lucid tree]

next, we try to find b and c

[lucid tree]

it shouldnt be equal to 0 always btw

[prime cosmos]

3n^2 + bn + c = 10

[lucid tree]

yes but thats ONLY when n=1

[prime cosmos]

3n^2 +bn + c = 21

[lucid tree]

yupss

and thats when n=2

[prime cosmos]

and so on

so what do i do next

[lucid tree]

well, u have 2 variables so u only need 2 equations

[lucid tree]

simplify this for n=2

in terms of b and c

[prime cosmos]

3(2)^2 + bn + c = 21

[lucid tree]

b also has a n term

[prime cosmos]

12+ bn + c =21

is b not supposed to have a n term

[lucid tree]

it is

so bn = 2b

for n=2

right?

[prime cosmos]

so this is right

as a simplified version

[lucid tree]

its supposed to have an n term

[lucid tree]

this

[prime cosmos]

wdym

[lucid tree]

is right

BUT

u didnt evaluate it for bn

whats bn if n=2?

[prime cosmos]

2b?

[lucid tree]

yess

[prime cosmos]

soo

[lucid tree]

jus a side question not related, whats bn if n=19

[prime cosmos]

12 + 2b + c = 21

[prime cosmos]

19b

[lucid tree]

yup

u got it

[prime cosmos]

12 + 2b + c = 21

[lucid tree]

thats ur second equation

[prime cosmos]

how many do i do

[lucid tree]

12+2b+c=21
3+b+c=10

[lucid tree]

2

[prime cosmos]

only two?

alwasy

[lucid tree]

yes

for quadratic sequences, yes

[lucid tree]

have u solved simultaneous equations before?

[prime cosmos]

yes

[lucid tree]

solve for b and c then from those equations

[prime cosmos]

ok wait a sec

ok

so i substitute n for 1 as my first equation

then n for 2 as my second equation

[lucid tree]

yup

[prime cosmos]

1st EQUATION

[lucid tree]

yeah

[prime cosmos]

2nd EQUATION

[lucid tree]

yeah

[prime cosmos]

SIMULATANEOUS EQUATION

i don't get how this answers the question tho

[lucid tree]

u need a general sequence an^2+bn+c

[lucid tree]

now find c

[lucid tree]

eh how did u get 12 on the RHS

in the second line

ur c value should be 5

16+c=21

[prime cosmos]

oh sorry mistake

A

[lucid tree]

so now u have 3n^2+2n+5=ur sequence

thats it

u r done

[prime cosmos]

no way

is that right

[lucid tree]

yups

try inputting n=3 and see what u get

[prime cosmos]

wdym?

[lucid tree]

in this equation

sub in n=3

[prime cosmos]

=38

Ohh

[lucid tree]

see

[prime cosmos]

i see how it works now

[lucid tree]

yups

[prime cosmos]

so how come i only need to use the first two equations

[lucid tree]

u can use any 2 equations

[prime cosmos]

like how come i don't substitue 3=n

[lucid tree]

if u have enough data, u could use n=3 and n=5

or n=1 or n=2

[prime cosmos]

oh i see

[lucid tree]

but u only need 2 equations, thats whats important

[prime cosmos]

but its simple to do just n=1 and n=2

like its easier

[lucid tree]

yups

[prime cosmos]

wow

thanks so much brop

[lucid tree]

np

[prime cosmos]

u really helped me out

i appreciate it

how many marks do u think this question would be

[lucid tree]

4 marks

[prime cosmos]

because theres alot more working out than i expected

ok cooll

thanks alot again

have a good day

[lucid tree]

u too!

.closed

uh

[prime cosmos]

.close

[lucid tree]

can u type .closed

[final saddle]

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[prime cosmos]

i gotchu

[lucid tree]

okay thanks

[final saddle]

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[worn skiff]

What did I do wrong here? Or how do I solve it correctly?

[barren hound]

i can't quite tell but does it help to point out that $\f{\sqrt2}2 = \f1{\sqrt2}$?

[soft zealot]

hayley!

[worn skiff]

I think I see my mistake. it should be |sin(x/2)| and then sign(...) and because of cos being alternately it's -1/√2 for 0^- and 1/√2 for 0^+ right?

And yes it helped, that you point that out haha

Thanks

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[plucky mountain]

I need help with this maths problem:

[hearty vale]

time = distance/speed

[plucky mountain]

I tried working it out with that formula in mind but I was still stumped on this question for half an hour

[hearty vale]

you have two equations

one for going to his house

one for coming back

and also two unknows, the distance and the time

so u can find both variables out by isolating one variable and plugging into the other equation

[plucky mountain]

What are the two unknown?

[hearty vale]

the distance and the time

[plucky mountain]

Thanks

[final saddle]

@plucky mountain Has your question been resolved?

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[quasi plinth]

.close

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[honest gorge]

so i have been doing this problem and i get it for the most part but my answer is slightly different because of a negative

i found this on photo math so i am just curious as to why it adds a negative in this case

for inside the integral

any help is appreciated thank you

[formal trail]

when you make the substitution t = cos(x), what is the derivative of that?

[honest gorge]

im dumb

nice

ok yea

alright yea thanks

i was thisahgoiehg

ahewioaghewiuoghjwa

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[half forum]

qs

solution

how do I find laplace transform of 2u dot + u

[tulip coyote]

Linearity and the way that Laplace transforms act on derivatives, no?

[half forum]

like I am not sure how to get the laplace transform

of the lhs

[tulip coyote]

"the same way you transformed the right hand side" [in particular, u dot gets transformed to sL(u) - u(0)]

Notice you have u(0) = 0 (wait is u a given function here?)

[half forum]

I don't get what u mean

[tulip coyote]

How did you do this?

[half forum]

just laplace transform

wdym

[tulip coyote]

The same applies to $\mathcal{L}(2\dot{u} + u) = 2\mathcal{L}(\dot{u}) + \mathcal{L}(u)$

[soft zealot]

@tulip coyote

[tulip coyote]

Do the exact same steps that you did with y, but replace y with u

You almost certainly should be happy that $\mathcal{L}(\dot{u}) = s\mathcal{L}(u) - u(0)$ as you would if it were $y$

[soft zealot]

@tulip coyote

[half forum]

so laplace transform of a unit step function is 1/s

correcto ?