help-49

[mint ravine]

but idk what to put for X, so that none of the cards have the same color

I thought 4 choose 3 would work (cause then you would have 1 of each color and you have to choose 3)

but that was wrong so idk what to do now

<@&286206848099549185>

[woeful turret]

ping after 15 minutes bro

[mint ravine]

sorry yoda

[woeful turret]

but first choose the colour which appears twice

what would that be

[mint ravine]

it could be any of the four colors

[woeful turret]

yes

so out of 4 how do u choose 1 colour

[mint ravine]

4 choose 1

[woeful turret]

yes

now that we have chosen a colour

we need to choose 2 numbers

how would we do that

[mint ravine]

can you specify what type of 2 numbers you are talking about

like for complementary counting or what

oh nvm

misunderstood it mb

[woeful turret]

from the colour we have chosen we need to choose 2 numbers

[mint ravine]

7 choose 2

[woeful turret]

correct

now choose a different colour from the remaining colours

how would u do that

[mint ravine]

3 choose 1

[woeful turret]

and now we have one card remaining to choose

can u tell how would we do that

also is this an ongoing test?

[mint ravine]

no

[woeful turret]

okay 👍

[woeful turret]

^

[mint ravine]

do we multiply them

[woeful turret]

we will do that at the end yes

but we have one step remaining

we have covered posibilities for the first 2 cards of same colour

and we have chosen the colour of the last card

but now it can be any number

how do we choose it

[mint ravine]

7 choose 1

[mint ravine]

7 ways to choose it

c(7,1)

[cedar pawn]

I'll do this by a different method and see if I get the same answer

[woeful turret]

yes

now multiply everything we have got

[mint ravine]

so 4 times 7 choose 2 times 3 times 7

[woeful turret]

uh

[mint ravine]

i cant use the * symbol when writing numbers for some reason

[woeful turret]

no

[woeful turret]

oh wait

yes

[cedar pawn]

Discord uses Markdown for formatting, and * is formatting for italics. If you want to use the symbol, you need to use a \ before it

[mint ravine]

oh ok thanks

.close

[midnight plank]

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[midnight plank]

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[twilit field]

would I be wrong in saying this has no eigenvalues or eigenvectors

[fallow scarab]

Did you prove it

[twilit field]

I was able to arrive at a contradiction \lambda x_1 = x_1 \implies \lambda =1; \lambda x_2 = 2x_2 \implies \lambda =2 and so on

[lavish venture]

why not x_1 = 0?

[twilit field]

though now that I think of it, this means the eigenvectors can only have one non-zero entry is what this means

[lavish venture]

yea

[twilit field]

got it

[lavish venture]

so what do you think the eigenvalues are then

[twilit field]

1,2,\dots n

the subsapces will be lines in F^n

cool

thanks

.close

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[molten bay]

x tends to infinity but n and x

Two variables?

[nova yoke]

i guess n is some fixed positive integer?

[midnight plank]

@molten bay Has your question been resolved?

[molten bay]

I see so what will be the approach?

Takin log?

[fallen sparrow]

Yes

[last slate]

Is the answer 1/n ?

[midnight plank]

@molten bay Has your question been resolved?

[gaunt nimbus]

My guess is

Consider u = 1/x

Do ln of the entire thing next

Can define f(u) = ln(sum from k=0 to k=n k^x)

Then it should be definition of derivative?

Not sure tho

[desert siren]

Is this from the integration chapter thingy where you take r/n as x and 1/n as dx? Or am i wrong

[midnight plank]

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[gilded bison]

Ti

[gilded bison]

5?

correct?

[gaunt nimbus]

,rccw

[gaunt nimbus]

,w -8000 -(800030.035) + 8000(1.035)^3

[grand pond]

Results provided by WolframAlpha

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

Requested by nuclearinteraction

[gaunt nimbus]

How did u get that?

[midnight plank]

@gilded bison Has your question been resolved?

[gilded bison]

i dont know much finical math

will i get this formula in test

[gaunt nimbus]

Do u know

[gilded bison]

I think i get finical formulas

[gaunt nimbus]

Formula for each form of interests

[gilded bison]

But can u show me the formula

[gaunt nimbus]

U can google it

[gilded bison]

whats it called?

[gaunt nimbus]

this is for compound interest

simlpe interest is this

[ A = P + Prt]

i think?

[grand pond]

k

[midnight plank]

@gilded bison Has your question been resolved?

[craggy grotto]

😮

COMPOUND INTEREST

I LOVE COMPOUND INTEREST!

Work out the Compound interest first

And then work out the simple interest

And subtract them from each other to get th Ans

[gilded bison]

What is the principal value?

I did this Shi last term

[craggy grotto]

The initial amount you deposit

So for example

I wanna deposit $800 to the bank

And they will give me interest for trusting the bank

It’s an easy way to gain money while saving money.

[gilded bison]

oh yeah i mixed that up with A

so lemme try it

using COMPOUND interest formula

[midnight plank]

Channel closed

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[gilded bison]

.reopen

[midnight plank]

✅

[gilded bison]

good

wait whats n?

@craggy grotto

would it just be 1

$8000(1+3.5/1)^1^3$

$8000(1+\frac{3.5}{1})^1^3$

[grand pond]

coehen1231
Compile Error! Click the reaction for more information.
(You may edit your message to recompile.)

[gilded bison]

$8000(1+\frac{0.035}{1})^1^\times^3$

there ya go ya big dumpa

[craggy grotto]

Did you get it correct?

[grand pond]

coehen1231
Compile Error! Click the reaction for more information.
(You may edit your message to recompile.)

[gilded bison]

im not sure have i?

$A=8000+8000\times0.035\times3$

[grand pond]

coehen1231

[gilded bison]

=8869.743

[gilded bison]

=8840

difference=

29.743

Seems like i have

,w $A=8000+8000\times0.035\times3$

[grand pond]

Results provided by WolframAlpha

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

Requested by coehen1231

[gilded bison]

,w $8000(1+\frac{0.035}{1})^1^\times^3$

[grand pond]

Wolfram Alpha doesn't understand your query!
Perhaps try rephrasing your question?
Click here to refine your query online

Requested by coehen1231

[gilded bison]

,w $8000(1+\frac{0.035}{1})^1^\times^3

[grand pond]

Wolfram Alpha doesn't understand your query!
Perhaps try rephrasing your question?
Click here to refine your query online

Requested by coehen1231

[midnight plank]

@gilded bison Has your question been resolved?

[midnight plank]

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[twilit field]

could someone explain the conclusion ( highlighted ) here

[twilit field]

nvm, I think I got it

.close

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[worldly pine]

answer is 27 years

[worldly pine]

.close

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[dire stag]

How is it 9 pi

[kindred wraith]

what would radius of circle be?

[night hawk]

What you drew is a square inscribed in a circle.

[kindred wraith]

ye just noticed

[night hawk]

Are you able to solve the question now?

[kindred wraith]

its not my question

[dire stag]

Yeah like how would you even justify 6 as the radius

[kindred wraith]

ye this the diagram

[dire stag]

It's not my paper lol

[dire stag]

I googled ts

[kindred wraith]

you could

but that later

[kindred wraith]

now see here

diameter of circle and side of square have anything in common?

[dire stag]

Btw diameter is a line that passes through the center of the circle and touches the circumference of the circle twice right

[kindred wraith]

yep

longest line which can be made inside a circle

[kindred wraith]

notice anything?

need a hint?

[kindred wraith]

align the diameter with the side of the square

[dire stag]

I mean ik that if u square 2 sides of a square and add them you get the diagonal of the circle

[kindred wraith]

that for when its the square on the inside

this time its the circle on the inside

wait a min lemme make the diagram

[dire stag]

What why?

[kindred wraith]

youll see once u see diagram

[dire stag]

I think I need a scale and and compuss for this 😭

This is crazy dude

[kindred wraith]

its very rough dont mind

[dire stag]

If a circle is inscribed in a square then it's diameter

Whoaaaassa

[kindred wraith]

now you see

[dire stag]

Ur the goat

Damn man 😂

[kindred wraith]

see how the side of square and diameter look same same

[dire stag]

Ts was Soo easy 😭 and obvious

[kindred wraith]

ye

that why u normally make a diagram first

before u attempt

[dire stag]

Tyy

[kindred wraith]

this when it the square on the inside

the opposite corners of the square make diameter

[kindred wraith]

but lets go back to this

[dire stag]

I solved it

[kindred wraith]

got it?

whats the answer

[dire stag]

Yes

9 pi

[kindred wraith]

yep

for questions like this first make the diagram then attempt

even is its a rough diagram it make this hella easier

[dire stag]

Real

[kindred wraith]

you got any more doubt?

[dire stag]

Nahhh

[kindred wraith]

ight

ima go then

[dire stag]

I didn't know it's gonna be this fast tbh

I thought if I create a post

Then id have to wait for hours 😭

That's why I've never done ts

[kindred wraith]

it dpends
i saw so i answer quickly

but for question which arnt collage level it should take like 10 mins max

oh yea @dire stag after your question done you type ".close"

[dire stag]

Mb

.close

[midnight plank]

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[empty ivy]

I'm so confused, what did I do wrong here for part c

[bright holly]

You did not evaluate the value of the exponential integral at t=0, it's not zero

[midnight plank]

@empty ivy Has your question been resolved?

[empty ivy]

Huh wdym

[bright holly]

What's e^(-kt) evaluated at t=0?

[midnight plank]

@empty ivy Has your question been resolved?

[empty ivy]

Ohh

Thank you!

.close

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[molten bay]

Square root of 1/(1-x)^2

[molten bay]

Is it correct to say 1/(1-x)

Or anything condition should be there?

[runic hamlet]

|1/(1-x)|

[wild eagle]

no

[chilly cobalt]

its 1/|1-x|

[molten bay]

I am thinking it has to be D

[wild eagle]

u dont ned in numerator

[molten bay]

But given answer is b

[runic hamlet]

and?

putting it around the whole expression isnt wrong

well wlog everything is always positive in india

[wild eagle]

I didnt say its wrong

I said u dont need it

[molten bay]

Wlog?

[runic hamlet]

without loss of generality

[molten bay]

Which option will be correct

[runic hamlet]

d

[molten bay]

Yes

They have to mention the absolute sign

[runic hamlet]

but well i've seen it often enough for these questions that people just ignore that things could be zero or negative

its not like this sum even converges always

[molten bay]

You meant b is correct

[bright holly]

tbh the actual exams specify all the details and don't leave anything open to interpretation or they give the question as bonus to everyone

[chrome laurel]

agp

ITS

AGP

so yeh

b is correct

not d

@molten bay u know how to do it

AGP problems

[subtle zinc]

what

[chrome laurel]

arthematic geometric progession

AGP

its combination of AP and GP

[wild eagle]

?

[chrome laurel]

its a series of both ap and gp

[runic hamlet]

we all know what it is

[chrome laurel]

i got b as answer

[runic hamlet]

that doesnt mean you can just ignore those details

[chrome laurel]

but

its a standard model of agp

they have to mention that x lies btw 0 and 1

[runic hamlet]

but they didnt

so it fails

[chrome laurel]

yeh

but u have to assume

[molten bay]

I know AGP but we can do it directly it is just derivative of series (1-x)^-1

[runic hamlet]

this is the whole thing we meant

no you dont have to assume

[chrome laurel]

if sum tends to infinity

[runic hamlet]

(well thats how the agp formula is derived)

[chrome laurel]

dunno

if x is less than 1

answer is b

if x is more than one answer is d

[runic hamlet]

and therefore because x was unspecified the answer is d

[chrome laurel]

is this given in a test

[molten bay]

Given answer is B

So surely it has to be |x|<1

[runic hamlet]

wel at least we dont have to worry about the expression being negative

its either positive or infinity

[midnight plank]

@molten bay Has your question been resolved?

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[midnight plank]

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[open bolt]

how do you do this sat problem I’m really confused

[fallen sparrow]

This is a test?

[open bolt]

kind of it’s a psat

[fallen sparrow]

Is it going on

[open bolt]

no it ended

[fallen sparrow]

Ok

What have you tried

[open bolt]

I can show u what I did

[fallen sparrow]

Show

[open bolt]

so I put it in Desmos when I was taking the exam and I found the two points it intersected w the line x

[fallen sparrow]

Which are?

[open bolt]

sry I’m opening desmos again

[fallen sparrow]

Wait

Before you do that

[open bolt]

yes!

[fallen sparrow]

Think about solving the problem in an easy way

Become "LAZY"

[open bolt]

wdym 😭😭

desmos is the laziest way right

[fallen sparrow]

Divide the quadratic by 3

[fallen sparrow]

Nooooo

[open bolt]

OHHH

wait

so 3(x^2-6x-5)??

[fallen sparrow]

Ys

= 0

So x^2 - 6x - 5 = 0

[open bolt]

oh yes

so

[fallen sparrow]

Also were you allowed to use desmos during test?

[open bolt]

it comes with the desmos calculator bc it’s psat

I think?

[fallen sparrow]

Oh I see

I didn't know that

Anyways

Now you understand it could've been solved in a much easier way

[open bolt]

I think it recently became digital

maybe it was different when u took it? if u did?

[fallen sparrow]

I didn't

[ornate bone]

yes

[open bolt]

ohh okay

but I’m still kind of confused

[fallen sparrow]

On what

[open bolt]

so like how do I simplify that

sorry

[fallen sparrow]

You got x^2 - 6x - 5 = 0

Right?

[open bolt]

yeah

[fallen sparrow]

So transpose 5 to the right side

[open bolt]

what does transpose mean

OHH

like move it

right

[fallen sparrow]

Uh okay forgrt that add 5 to both sides

[fallen sparrow]

Yea

[open bolt]

oh okay

[fallen sparrow]

So x^2 - 6x = 5

[open bolt]

so now I would do x(x-6)?

[fallen sparrow]

There you have it

No

[open bolt]

oh

[fallen sparrow]

Why would you do it

[open bolt]

are u sure

[fallen sparrow]

You've got the answer now

[open bolt]

im trying to find X right 😭

[fallen sparrow]

No aren't you trhna find what is x^2 - 6x

[open bolt]

OHHHH

I thought I was supposed to find X and then do that

sorry tysm!!!

can I ask u another question

[fallen sparrow]

Sure

[ornate bone]

you can use a cas calculator

well u could

probably not in august

[open bolt]

ty!!!

how do u do this

[open bolt]

what is that

[ornate bone]

well the graphing calculators

[open bolt]

ohhh so like desmos

[ornate bone]

u can do it by hand or use desmos regression

[open bolt]

what’s desmos regression

[ornate bone]

desmos is a numerical calculator of sorts

[fallen sparrow]

What have you tried

[ornate bone]

cas gives you an exact answer

[open bolt]

oh so

[ornate bone]

finding the best fit exponential given that it passes through those two points

[open bolt]

wait how do u do that

🤯🤯🤯

[open bolt]

I forget but I can try again

[open bolt]

It js said I was wrong

[open bolt]

what do those letters mean

and how did u write those equations

sorry I’ve never seen this before

[ornate bone]

i defined a new function g(x) = f(x) - 15

then the next line is essentially telling desmos to "solve a system of equations" (not really but that's the case on the SAT)

1. g(0) = -99/7 (given in the question)

2. ab = 65/7

[open bolt]

ohhh

okay

[cedar vector]

help me

[open bolt]

me?

idk if I can but I can try

???

[ornate bone]

this channel is occupied

get your own

[open bolt]

hi sorry

I just have a rly quick question

how did u get -99/7

[cedar vector]

how to

[ornate bone]

g(x) passes through (0,-99/7)

it's given in the second sentence

[ornate bone]

#help-43

[open bolt]

yes but like how did they end up together if that makes sense

im sorry

[ornate bone]

maybe u should learn how to work with regressions first

[ornate bone]

g(0) = -99/7

so they take the first place in each list

[open bolt]

oh okay do u have any recommendations to how to learn

[ornate bone]

[input list] ~ [output list]

[open bolt]

ohh okay

[ornate bone]

hmm good question

[open bolt]

how did u know the input is like g(0)

[ornate bone]

what else would you put?

[open bolt]

bc it’s not in the equation

what is g

[ornate bone]

Do you agree that we know these two things:

1. g(0) = -99/7
2. ab = 65/7

[ornate bone]

f(x) - 15

i just defined a new function

u don't need to

[open bolt]

yes but I didn’t know g was 0?

oh okay

[ornate bone]

f(0) - 15 = -99/7

that's what sentence 2 says

[open bolt]

yes

[ornate bone]

g(0) = f(0) - 15 = -99/7

thus g(0) = -99/7

[open bolt]

ohhh

[ornate bone]

[open bolt]

so I could technically use any letter

why is it g?

sorry

is that like needed for desmos

[ornate bone]

better?

[open bolt]

ohhh

yes sorry that makes sense!

I think I js need to learn regressions

do u have any suggestions for where to learn this

[ornate bone]

I mean you can certainly do the math

But a lot of people don't actually know the math

and still get 750ish with desmos lol

[open bolt]

ohh okay thats me

like idk the math

but I also don’t know the desmos

[ornate bone]

I mean it's not a substitute for not learning the math

U should still learn it

But yeah

[open bolt]

yeah 😭

[ornate bone]

If your goal is like a 700 then maybe not that much

[open bolt]

can I show u what I got

[ornate bone]

for what?

[open bolt]

the psat

[ornate bone]

Oh

Sure

[open bolt]

is that bad

😭😭😭

[ornate bone]

😭

idk depends on your goal

On the SAT

[open bolt]

what should my goal be

sorry

[ornate bone]

you can get 600 with desmos alone

tbh

[ornate bone]

Idk

That depends on you

[open bolt]

okay ty 😭

do u have any recommendations for how I can improve

[ornate bone]

Did you take a ptest?

[open bolt]

what’s test

[ornate bone]

ptest = practice test

[open bolt]

ohhh yes

[ornate bone]

for what?

math?

[open bolt]

yeah

like what math should I learn

[ornate bone]

U should find your goal score first 😭

[ornate bone]

What are you the weakest at?

[open bolt]

advanced math i think?

I just took precalc this yr so maybe I need to learn calc?

[ornate bone]

yeah "advanced math" is just the constant questions

technically just algebra

[open bolt]

oh 😭

[ornate bone]

no

everything is below precalc

or precalc at best

first few chapters of precalc

[open bolt]

oh 😟

okay

[ornate bone]

even without precalc u should be able to get 700+

[open bolt]

okay 😭

what should I do if I want to get a 700

I thought I was pretty good at algebra ngl

[ornate bone]

when's your exam?

[open bolt]

I haven’t scheduled it yet!

this was my psat from my school

[ornate bone]

When do you plan to take it?

Or rather how long do you have to prep

[open bolt]

when do u recommend

[ornate bone]

😭

Okay

take a practice test first

[open bolt]

sorry 😭

I js did

I’ll take another one

[ornate bone]

Okay show the split

[open bolt]

what’s that

like my other one?

[ornate bone]

I mean what did you get on your practice test

[ornate bone]

That's pretty good

[open bolt]

really 😭😭😭

[ornate bone]

I mean your english score is really high

and if this is your score without prep

then it's pretty good

[open bolt]

yeah I thought it wasn’t too hard

thank you 😭😭😭

[ornate bone]

Wait, have you taken an official practice test?

Like the one offered by collegeboard?

[open bolt]

this?

like my school

[ornate bone]

It's not from bluebook tho, is it?

[open bolt]

no

it’s from

uhh

franklin yard?

[ornate bone]

Okay cool cool

Why don't you take an official practice test

From bluebook

[open bolt]

how do I do that

[ornate bone]

It's free

[open bolt]

okay!

[ornate bone]

https://bluebook.collegeboard.org

Make an account

and then search for practice test

Take practice test 4 or something

[open bolt]

ohhh okay tyty!!

[ornate bone]

The math is a bit easier than the real deal

but it's still an official source

so it's the best one out there

it'll give you a good baseline of where you're truly at

[open bolt]

okay tysm!!!!

I’ll take it tmr!!

💗💗💗

[ornate bone]

Sure, gl

[open bolt]

thank you!!!!

did u already did the sat?

[ornate bone]

Yes

[open bolt]

do u have any other tips!!!

I rlly appreciate it 😭😭😭

[ornate bone]

Not really 😭

But yeah you should probably do a bunch of questions with regression

There's a whole questionbank

[open bolt]

okay 😭😭😭

[ornate bone]

So you could start with those

[open bolt]

what’s a question bank

[ornate bone]

Then there's bluebook+

[open bolt]

what’s that

sorry 😭

[ornate bone]

a place where you find sat questions (official ones)

[open bolt]

OHHH

WOW

[ornate bone]

https://oneprep.xyz

[open bolt]

is that legal

[ornate bone]

I guess they compiled it here

But again it's accessible for free

[ornate bone]

the question bank? Yes

[open bolt]

okay tyty !!!!!

[ornate bone]

bluebook? yes

[open bolt]

thank u!!

[ornate bone]

bluebook+? maybe not lol cuz those are past tests

[open bolt]

so can I js like memorize these

[ornate bone]

No

[open bolt]

oh okay 😭😭😭

[ornate bone]

They're practice questions issued by collegeboard

But since they're the test makers

It's going to be of a higher quality than something you'd find in third party resources

[open bolt]

ohhh okay ty!!

that makes sense!!

I really appreciate the resources!

[ornate bone]

Lol sure

np

[open bolt]

have an amazing day!!!!

I’ll study rlly hard!!!

[ornate bone]

Mhmm gl

[open bolt]

[ornate bone]

[midnight plank]

@open bolt Has your question been resolved?

[midnight plank]

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[keen harbor]

I'm writing some notes from a book. In these notes, I'm trying to create an anti-example for a given definition.
I'm wondering if what I wrote is a correct way to noting what I'm trying to say.

[keen harbor]

In the underlined part, I'm trying to say something like "3 is not a divisor for 7 because there is no integer that multiplies 3 to give 7".

Want to make sure that notation isn't busted.

[crystal plover]

b3 is not really nice to read

b⋅3 is better

but everything else looks okay

Also doesn't one usually say “non-example”? Though am not a native English speaker

[keen harbor]

yeah, though I personally not amorous of that term.

Thanks for the feedback.

.close

[midnight plank]

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[narrow edge]

why does the arrangement of keys on a keychain stay t he same whhen its flipped over

also why is it mutiplied by two

[stray epoch]

why would it be any different

flipping a keychain over doesn’t affect the order of the keys

[midnight plank]

@narrow edge Has your question been resolved?

[narrow edge]

(check the last paragraph) reflection matters for a circle then shouldnt it also matter for a keychain which essentially has the same diagram?

<@&286206848099549185> <@&286206848099549185>

<@&286206848099549185> <@&286206848099549185>

[stray epoch]

I suppose it is because thinking about a keychain as a 3d object, when the keychain is flipped upside down you consider the perspectives of the keys to be flipped as well. Just a guess.

[narrow edge]

so the keys arent a b and c?

<@&286206848099549185>

<@&286206848099549185>

<@&286206848099549185>

[fallow scarab]

Good thing helpers ping doesn't do anything

[narrow edge]

wait actually?

[fallow steppe]

no

[narrow edge]

i am waiting since 45 minutes smh

[fallow steppe]

i keep getting pinged here

[fallow scarab]

You can remove your helpers role

[fallow steppe]

all the helpers keep getting pinged here

[fallow steppe]

nah i’m fine w it

but let me see if i could help

okay so i’m not too good at this rn

but if i had to assume it’s because a circle has no sides

and because it had no sides and all the points are spread across recently

evenly*

spinning it doesn’t change the angles or distance between the points

[narrow edge]

yea i get the rotations part

i dont understand reflections

[fallow steppe]

let me see

[narrow edge]

the reflection of the points on a circle gives a different arrangement but when it is the keys of a keychain the arrangement does not chaange

[fallow steppe]

the keys of a keychain aren’t circled

circles

so

here’s how the reflections work

so you know rotating doesn’t change the order they are in

at it’s a b c

no matter the way it’s spun

however when you are reflecting something the points change

say

a
b c

spin it

b
a c

the numbers next to a are the same (along with the other points)

now we will reflect it

a
b c

turns to

a
c b

the order of the letters are different than before

does that make sense

[narrow edge]

shouldnt we get bca after spinning?

[fallow steppe]

depends on how you read it

i was going top right left

[narrow edge]

yea

[fallow steppe]

okay

[narrow edge]

so why do circle have a different arrangement when they are flipped and keychains dont when they are essentially the same thing?

[narrow edge]

.?

[midnight plank]

@narrow edge Has your question been resolved?

[fallow steppe]

so sorry

[fallow steppe]

honstly

i have no idea

but if i had to assume

it’s because a circle is a circle

and keychains if you flip them look different

[midnight plank]

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[molten bay]

I want to understand this root process

[hallow hazel]

Synthetic division

By observing the equation you can clearly see that 1 is one of its zeroes

That means x-1 is a factor of it

Synthetic division is essentially a short hand method of dividing the original expression by this factor

[midnight plank]

@molten bay Has your question been resolved?

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[tiny bolt]

Could someone help me solve this integral step by step? The solution on symbolab is super messy and hard to grasp

[lyric charm]

can we see what symbolab is suggesting to do

[tiny bolt]

Do you want me to like type it out or screenshots

[lyric charm]

screenshots are better

[tiny bolt]

although the final answer isnt fully captured

[lethal owl]

So the first step here of u substitution isn't super clear

U = r^{1/4}?

Du = (1/4) U^(-3/4)

[tiny bolt]

Unfortunately I don't have symbolab pro

There's barely any clues

It's direcf

[lethal owl]

Ok let's try that
u = r^(1/4)

r=u^4

dr = 4u^3

Yeah that's clear now

$$\int\frac1{r^{\frac14}(r+1)}dr$$

[grand pond]

Sherif Player

[lyric charm]

dr = 4u^3 du btw

let's not drop pieces of notation

but it looks like yeah that's the only way forward regardless

and after that it's going to be quite ugly

[tiny bolt]

Then it uses partial fraction

But at one part it just directly skips

[lethal owl]

$$\int\frac1{u(u^4+1)}4u^3du$$

[grand pond]

Sherif Player

[tiny bolt]

With no explanation at all, just the solution

[lethal owl]

Yeah

[lethal owl]

The problem is that thing doesn't have real solutions
It can be factored to 2 quadratic equation
But no more than that

The denominator

So what was in my mind is
trig substitution

[tiny bolt]

i see now

[lethal owl]

$$\int\frac{4u^2}{(u^4+1)}du$$

[grand pond]

Sherif Player

[tiny bolt]

It directly uses the standard integral of x/(ax² + bx + c) dx

[lethal owl]

Yeah he used the formula
$$1 + u^4 = 8(u^2-\sqrt{2}u+1)(u^2+\sqrt{2}u+1)$$

[grand pond]

Sherif Player

[tiny bolt]

I see

My query has been solved

.close

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[chilly cobalt]

Let a,b,c be disinct reals >= 1. Prove that
|a-b| + |b-c| + |c-a| + 6 <= 2sqrt(2)(a+b+c)

[chilly cobalt]

wlog let a > b >c >= 1 then i have
a -c + 3 <= sqrt(2)(a + b + c)

so i have to prove sqrt(2)(a+b+c) + c-a >= 3

stuck here💔

[viral dagger]

well all of them are >=1 so sqrt(2)(a+a+c)+c-a>sqrt(2)(a+2c)+c-a>(a+2c)+c-a>=(a+2)+1-a=3

[chilly cobalt]

oh

wait but if i do that then i dont have equality

?

[viral dagger]

wait nvm

[viral dagger]

this should be fine?

but why is the inequality so loose

[chilly cobalt]

ehh

seems a bit sketchy

we still dont have equality case thouhg

[viral dagger]

idk maybe there isnt an equality case lol

[chilly cobalt]

hmm

oh wait

equality case happens only when a =b =c which is impossible

sighh

rahhh ty

.close

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[viral dagger]

doesent touch

[midnight plank]

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[chilly cobalt]

find primes p such that 4p^2 + 1 and 6p^2 + 1 are also prime.

[chilly cobalt]

how do i start?? surely its not bash right

[midnight plank]

@chilly cobalt Has your question been resolved?

[tawdry laurel]

hmmmm interesting question

[tawdry ermine]

Primes above 5 are usually denoted in the form of 5k +- 1 or 5k +- 2. Take note of that

Well, using "modulo 4" states that a number should be divisble by 4

[tawdry laurel]

yes but i was referring to the remainder until i realised it failed

[tawdry ermine]

For both 4p^2+1 and 6p^2+1 to be prime, it's modulo should not have a remainder

My advice for you is to try plug in the first 3 prime numbers into both expressions and see if they're prime numbers

[chilly cobalt]

p = 5 works i think?

[tawdry ermine]

Yes

So now try working it out if p =/= 5 using modulo 5

[chilly cobalt]

what does =/= mean😭

[tawdry ermine]

p is not equal to 5

[chilly cobalt]

oh

aight

p^2 mod 5 is either 0, 1, or 4 with = 0 only when p = 5

and then case check leads us to p = 5?

[tawdry ermine]

Yes

There you go

[chilly cobalt]

goated!!!

tyty

[tawdry ermine]

It's my pleasure

[chilly cobalt]

so im done with the proof now?

[tawdry ermine]

Yep

Pretty much

[chilly cobalt]

tysmm!!

.close

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[woeful turret]

((x^2+1)2^(x-1))/x(x+1)

[woeful turret]

i have to do summation of this series from 1 to n

how can i split it into like a telescoping sum?

my friends told me that u can split it into 3 terms

[zealous schooner]

[ \sum_{r=1}^n \frac{(r^2+1)2^{r-1}}{r(r+1)} ]

[grand pond]

kheer257

[zealous schooner]

this is the sum you want?

[woeful turret]

yeah

thanks for the latex text

[zealous schooner]

yeah so you need to split this

into two terms

[woeful turret]

oh only 2?

[zealous schooner]

well I'm not sure

you want to split it into the form T_r - T_{r-1}

[woeful turret]

yeah

[zealous schooner]

so you likely want something of the form (...) * 2^r - (...) * 2^(r-1)

[woeful turret]

ok so i can write
1/x(x+1) as 1/x - 1/x+1

[woeful turret]

ok yeah

[zealous schooner]

The easiest way to do this without any guesswork is probably to perform partial fraction decomposition on the rational bit

[woeful turret]

oh ok so i have to convert
(r^2+1)/r(r+1) into a telescoping series

oh also

there were options in the question

and when my friends asked me i was like put n=2 and see which options match

and i got the correct answer as well

but they were like no ur not allowed to do that

[dreamy lichen]

thats def the easiest way (if you dont have to show your work and actually derive / prove the formula)

[zealous schooner]

not quite

perform the partial fractions first

[woeful turret]

ohh ok 👍

[woeful turret]

uhh

not sure how to do that exactly

i havent been taught that as a concept

[dreamy lichen]

seems like kheeri went offline

tbh i dont remember how exactly partial fractions are done either, so i use my own method which is kind of a guesswork

[fallen sparrow]

It's more kf a trial and error stuff

Assume Tr = br * 2^r

Tr - Tr-1 is what you wanna do

So solve for that

[woeful turret]

oh ok

ill come back to this question..thanks guys

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[chilly cobalt]

From the Image, C is the center of a circle with AB as a chord and E bisects AB (and H is C E H collinear, i drew it wrong) Prove that EF >= GH

[chilly cobalt]

i dont know where to begin i suck at geo😓

im assuming its like to find some similarities ?

[fallen sparrow]

You wanna do it through pure geometry or is coordinate fine?

[chilly cobalt]

pure geometry would be better

[dreamy lichen]

and D is arbitrary?

[chilly cobalt]

i think so yeah!

the problem only states these

[dreamy lichen]

whats H?

oh is it also arbitrary?

[chilly cobalt]

yehh

original img for reference

[dreamy lichen]

okay

oh you didnt mention one key detail

H lies on the bisector (radius)

without that fact its unsolvable

it still looks kinda weird though

[chilly cobalt]

OOPS

mb man

[dreamy lichen]

i feel like the claim is false

[desert siren]

I forgot to put 2

[dreamy lichen]

if we were to push D close to the horionztal diameter, it's pretty clear

[desert siren]

Nm its still false

Its not at the center

[chilly cobalt]

well

[dreamy lichen]

Push D towards horizontal diameter and do the same with AB

[chilly cobalt]

if D were on the diameter FD wouldnt pass E

[dreamy lichen]

push it close to it

[dreamy lichen]

if you do that, then EF will approach CA (i.e. one radius)

while GH will approach sqrt(2)*radius

[chilly cobalt]

hmm

damn

[hallow hazel]

I think h is not arbitrary

[dreamy lichen]

yeah, its not

but even then

[hallow hazel]

I think g is arbitrary

[dreamy lichen]

ill draw a pic...

[dreamy lichen]

if G is arbitrary then H is arbitrary as well

oh

do you mean like

[hallow hazel]

H depends on both

[dreamy lichen]

D,G,H dont lie on a line?

[chilly cobalt]

but like, the ineq should be correct? since this is like the pre-national olympiad test

[dreamy lichen]

certainly a poorly worded

unless you forgot to mention sth

[hallow hazel]

G is an arbitrary point on AB and H is the point of intersection of circumference ?

[chilly cobalt]

ive said everything mentioned😓

oh

maybe because this qns written from memory

theres this one part i have no idea what its abt

[dreamy lichen]

here, it should be clear that GH > EF

[hallow hazel]

If you do this assuming H is dependent on G, then they both equal each other

[dreamy lichen]

check this pic @hallow hazel

[subtle grove]

it can be prove via similair shapes right

[dreamy lichen]

it cant be proved because it false, unless i still misunderstand the problem

[chilly cobalt]

unless its some weird fact i dont know about law of sines?

[hallow hazel]

Your fixing H and drawing the line?

[dreamy lichen]

Well, where should H be?

Isnt H supposed to be all the way down?

such that C,E,H lie in a line

[hallow hazel]

Is that a condition?

[subtle grove]

H should be there

[dreamy lichen]

whats the source of the question?

[dreamy lichen]

no idea, guessing from the pic

[subtle grove]

think so

[hallow hazel]

It that’s a condition then the claim is false

[dreamy lichen]

even if it wasnt a condition, we would be left with arbitrary H

and it would still be false

[chilly cobalt]

darn🥀

[dreamy lichen]

i cant imagine an interpretation under which it would be true

[dreamy lichen]

The diagram is not even to scale >_< the C isnt even in the center

[dreamy lichen]

is there anything else? If so, can you send it?

Even if its in foreign language

[midnight plank]

@chilly cobalt Has your question been resolved?

[chilly cobalt]

translates to

Let C be the centre of a circle,
AB be a chord of that circle
Let E be the midpoint of AB
(law of sine EFH, EHG)
prove that EF >= GH

.close

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[drowsy wing]

can someone explain line 4 and 5

[drowsy wing]

i dont know how they rearranged to get that

[dreamy lichen]

rearranged what exactly

[dreamy lichen]

(1-e^(-x))^-1 gives you one y

so youre left with $y\cdot\left(-e^{x}\cdot\left(1-e^{x}\right)^{-1}\right)$

[grand pond]

MathIsAlwaysRight

[dreamy lichen]

with me on this?

well

$y\cdot\frac{-e^{x}}{1-e^{x}}$

[grand pond]

MathIsAlwaysRight

[dreamy lichen]

you can then change it to this

[drowsy wing]

should it not be -x

[drowsy wing]

for this

[dreamy lichen]

Yeah, right, good catch

$y\cdot\frac{-e^{-x}}{1-e^{-x}}$

[grand pond]

MathIsAlwaysRight

[dreamy lichen]

so we would be here

then we can add and subtract 1 in the numerator

$y\cdot\frac{-1+1-e^{-x}}{1-e^{-x}}$

[grand pond]

MathIsAlwaysRight

[dreamy lichen]

now we can split up the fraction

[drowsy wing]

yeh i see it now

[dreamy lichen]

$y\cdot\left(\frac{-1}{1-e^{-x}}+\frac{1-e^{-x}}{1-e^{-x}}\right)$

[grand pond]

MathIsAlwaysRight

[drowsy wing]

ight thanks

[dreamy lichen]

and it becomes y(1-y)

[drowsy wing]

.close

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[radiant lagoon]

Ok, so the only counter example I am able to think about is taking the geometric sum till the nth term

[radiant lagoon]

to show that C^1[0,1] to be incomplete

[lyric charm]

geometric sum of what

[radiant lagoon]

$f_n(x) = \sum_{i=0}^n (2x)^{i}$

[lyric charm]

\sum

$f_n(x) = \sum_{i=0}^n (2x)^i$ --- is that a Cauchy sequence in $(C^1[0,1], \nrm{\cdot}_{\infty})$?

[grand pond]

Ann

Xyrokryen

[radiant lagoon]

Oh wait, it isn't a Cauchy sequence