help-23

[terse atlas]

prove BxA^2 = (A^2)xB

basically that the matrices commute

that is the matrices A^2 and B, which are bot arbitrary square matrices over an arbitrary field, not Z_mod2 though

I think I veered very far off the path

[terse atlas]

I inferred 2 different equations here from the previous steps, not standard notation I know

[vital bison]

You know that BA = - AB

BA^2 = -A(-AB) ==> BA^2 = A^2B

[terse atlas]

Oh wow, once again I missed that I already proved something, thank you!!!

.solved

[safe radish]

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[terse atlas]

.reopen

[safe radish]

✅ Original question: #help-23 message

[terse atlas]

hold on, I don't see how you reachedBA^2 = -A(-AB) could you show how you expanded this?

just to make sure there isn't confusion on either end, I am trying to prove (A^2)xB = Bx(A^2)

[vital bison]

You derived that B^2A = - ABA

Now use this again

BA = - AB (the first equation)

B^2A = -A(-AB) = A^2B

[terse atlas]

got it!

for real this time

.solved

[safe radish]

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[modest holly]

How do i identify the turning points through a graph?

[burnt notch]

Depends on how accurately the graph is drawn or on the precision required

Apart from this, just recall the definition of turning point

[modest holly]

But thats where im having trouble

[floral yew]

dont draw the graph its not easy

[modest holly]

Is it 3?

[floral yew]

could be how did you solve it?

[modest holly]

graphed it

[floral yew]

oh well then yes

if a graphical calculator is allowed then use it for such questions

[modest holly]

alr thanks

[floral yew]

if not then find minima and maxima where the derivative changes its sign thats your turning point

[modest holly]

didnt learn derivs

[floral yew]

so you're expected to use a graphical calculator for such questions?

alr

[modest holly]

.close

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[errant shuttle]

hey guys is this snowflake sudoku solvable?

[safe radish]

Please don't occupy multiple help channels.

[quasi bison]

what are the rules

[errant shuttle]

Its normal sudoku + along the lines must be different numbers(diagnoal)

I have been trying for 5-6 hours now 🙁 (I got 3 errors but all of them were different)

[safe radish]

@errant shuttle Has your question been resolved?

[quasi bison]

#recreational-math might be a better place to post this

[desert juniper]

it seems to have, i'm almost done with it

it does

[safe radish]

@errant shuttle Has your question been resolved?

[safe radish]

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[dull coyote]

I used a rectangle so that I could box in region R .used f(4) since graph c touches it ,I then multiplied f(4) by 4 which gave 52/3

Then did 52/3 subtract integration

[vale vessel]

,r ccw

[dull coyote]

What does that mean

[vale vessel]

theres a bot command that rotates pictures i forgot which one it was

[north raven]

,rotate

[vale vessel]

thank you

where did 52/3 come from

youll need to show more work than just the picture i fear

[dull coyote]

Hmm okay ill put it real quick

[vale vessel]

your answer is correct but you need to show your work

[dull coyote]

I did the first part and got the answer, finding the region is what I'm struggling with because the mark scheme said its 37/9

[vale vessel]

okay, you could do it that way, but there's way easier of a way to do this

[dull coyote]

Oh okay what is it

[vale vessel]

so P is $(4, \frac{13}{3})$, right?

[flat frigate]

Mirror

[dull coyote]

Yh

[vale vessel]

if i told you to calculate the area of the gray region + the area of the white region, how would you do that?

(bounded on 0≤x≤4, since P's x-coord is 4)

[dull coyote]

I think I'd make a rectangle

Then subtract the integral of C from it

[vale vessel]

yes that

how would you find the area of just the white region?

[dull coyote]

Oh a triangle

[vale vessel]

try a different way

if i just gave you the equation

[dull coyote]

Not sure tbh

[vale vessel]

what does $\int_{0}^{4} \frac{13x-26}{6}, dx$ represent?

[flat frigate]

Mirror

[dull coyote]

Oh that's the equation of the line rearranged so I could find y when x=4

[vale vessel]

that is the equation of the line, but what would that integral give me?

[dull coyote]

The integral gave me 76/9

[vale vessel]

im not asking you to calculate it, just what would it represent?

[dull coyote]

Area under the curve of C

[vale vessel]

nope

it would give me the area under a curve, but not curve C

want another go?

[dull coyote]

Yes

[vale vessel]

have at it!

[dull coyote]

I'll give it a go rn

[vale vessel]

no no hang on

[dull coyote]

Graph looks like this no annotations

[vale vessel]

Right, and you've done well to slice a line down from P

[vale vessel]

You know the thing being integrated is the equation of line I, so what does the integral represent?

[dull coyote]

My mistake the integral is for the curve of C

Line L is the 13x-26/6

[vale vessel]

Okay. If we do $\int_{0}^{4} C , dx$ would that give us the right answer for the area of region $R$?

[flat frigate]

Mirror

[vale vessel]

don't actually calculate it, just tell me yes or no and why

[dull coyote]

I'd say no since it would be everything under C

Everything above the x axis and below c

[vale vessel]

Right. But we'll need that area under C for something. What do we need to not count?

[dull coyote]

To count the area after x=4

[vale vessel]

sure, but we can do that by putting a boundary at x=4, like you did in the definite integral. what else do we need to not count?

[dull coyote]

The like lower white area of line L between x=2 and x=4

[vale vessel]

right, and this integral would count that

how can we find the lower white area?

what we want to do is find (area under curve C) - (lower white area)

[dull coyote]

Make a triangle between x=2 and 4 ,use integral (4) -triangle

[vale vessel]

that would work here, but can you make an integral that would find that area?

[dull coyote]

Integral of Line L

[vale vessel]

bingo

so written all the way out, what's the integral you need to calculate?

[dull coyote]

Between 2 and 4

I need integral of Curve C between x=0 and 4 subtract integral of Line L between x=2 and x=4

[vale vessel]

yes! math time

[dull coyote]

Ahh I see now that so much easier

Oh my original didn't work because it include the area under C which we didn't want

[vale vessel]

yeah. i did the integrals wrong the first time and did the L integral between 0 and 4, so you're not the only one

(||37/9|| is correct, btw)

[dull coyote]

You were so patient, are you a teacher?

[dull coyote]

They're a bit tedious so I like to draw things out

[vale vessel]

thank you! i am not a teacher

(i do teach, but not as a profession)

[dull coyote]

Nice you're pretty good at it ,I see it says under grad ,isn't that above bachelors

[vale vessel]

i'm in my last year of undergrad

[dull coyote]

Hope you're liking it , thanks so much

[vale vessel]

of course! if this is sorted make sure to mark the thread as closed with .close

[dull coyote]

.close

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[dry timber]

ok guys its really basic i just need someone to tell me what kind of fonction makes a shape like this and maybe try to give me the function that seem to match to this image the most

[long oracle]

Is this for a test?

[golden raven]

my first thought would be something like -arctan(x) + pi/2

but its hard to tell

[vital bison]

Would a test's question be this vague?

[dry timber]

no its a home work

[long oracle]

We won’t give out answers for ur homework

[vital bison]

This could be a tanh, a sigmoid, arctan as you said, its really vague honestly

[long oracle]

We’re supposed to guide you towards it

!xy

[safe radish]

Please show the original problem, exactly as it was stated to you, with the entire original context. A picture or screenshot is best. If the original problem is not in English, then post it anyway! The additional context might still be helpful. Do your best to provide a translation.

[vital bison]

Do you not have more information

[dry timber]

if u want more info here

[long oracle]

Why do you want to find the function? The question is intentionally vague and wants you to use the graph

What happens to g(x) as x goes more and more to the left of the graph?

[dry timber]

because i want to understand what kinda of fonction give this sort of graph

[long oracle]

This would be an answer to that

[long oracle]

Btw I could very well define a function that looks like this, but differs outside the shown domain

[zealous ravine]

im about to start learning functions in school any tips

[dry timber]

sure that would be helpfull

[zealous ravine]

grade 9

[long oracle]

#❓how-to-get-help

[zealous ravine]

ok thanks

[long oracle]

Well they specify transcendental, so something like this would look like the graph and satisfy the conditions of g

However..

In this case the limit as x approaches -infty is 0

But one would believe from the graph alone that it would be 300

[safe radish]

@dry timber Has your question been resolved?

[dry timber]

ok thx

[safe radish]

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[stuck owl]

how do i find the bounds for this integral? i know the points of intersection are (-3,0) and (3,0), but i don't know how to proceed from there confidently

[safe radish]

@stuck owl Has your question been resolved?

[stuck owl]

.close

[safe radish]

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[fossil zephyr]

One message removed from a suspended account.

[safe radish]

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[proven oak]

I am having trouble understanding fourier approximation, any advice or help?

[vagrant ice]

oh, you want the value at $x = 0$ to match

[flat frigate]

south

[vagrant ice]

since you already know the coefficients of cos(x), cos(2x) .... are all 0

you also know that sin(x), sin(2x), ... at x = 0 all become 0

[proven oak]

im sorry i dont quite understand

[vagrant ice]

when x = 0, almost all the terms vanish to 0

[proven oak]

okay so a0 = -4/sqrt(2) ? when x is 0

[vagrant ice]

no, because that wouldn't make $-4 = a_0 \frac{1}{ \sqrt 2}$

[flat frigate]

south

[proven oak]

OH -4sqrt(2)

[vagrant ice]

yep!

,calc -4 sqrt(2)

[flat frigate]

Result:

-5.6568542494924

[vagrant ice]

so to recap, you can first find the Fourier series for g(x) = 2x

that's an odd function, so c_k = 0 for all the ks (sin is odd, cos is even)

then you just subtract 4 from that, but the Fourier series has a_0 / sqrt(2)

[vagrant ice]

which gives you this equation

[proven oak]

so for even function b_k would be 0 ?

[vagrant ice]

yep!

we're relying on the property that:

1. sums of odd functions are always odd
2. sums of even functions are always even

e.g, you can check what happens to f(-x) + g(-x)

[proven oak]

right thank you so much!

[vagrant ice]

no worries!

.close

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[boreal imp]

Hallo
For b(i) , how can I find the vector OF so as to find DF? Or is there another way to find vector DF🤔

[feral linden]

You are solving 6b(i)?

You can simply let F=s(OB)+(1-s)(OC) and DF•BC=0 will let you obtain this s

[safe radish]

@boreal imp Has your question been resolved?

[boreal imp]

Ahhhhh I’m guess I’m dumb 😅
BTW thx for your help

[safe radish]

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[verbal kayak]

<@&286206848099549185>

[river relic]

No bro its alg

I was being dumb 😭

.close

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[hazy harness]

can soemone help me study for my math exam

[hazy harness]

i have quadrilatarals, circles and statistics

please help me

[austere goblet]

hi, and welcome to the server! please share any questions you're stuck with and we'll look into it.

[hazy harness]

honestly i need help with a lot of stuff

i have quadrilatarals in 9th grade

im not understanding anything 😭

[limpid lodge]

u can send in some questions?

[hazy harness]

nvm

.close

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[orchid mirage]

equation of the line passing through -2 √2,-3 √3 and the origin

[magic junco]

Are they coordinates?

[near atlas]

do you know the two point form a straight line

[orchid mirage]

yeah

[orchid mirage]

yeah it’s above

origin is (0,0)

[near atlas]

The equation of a line passing through $(x_1,y_1)$ and $(x_2,y_2)$ is given by : \
$y-y_1=\frac{y_2-y_1}{x_2-x_1}(x-x_1)$

[flat frigate]

T&C

[orchid mirage]

oh thanks

how do I write the solution

in terms of y?

[near atlas]

well after subbing in the suitable values of x1, x2, y1, y2 and simplifying. its easy to get an equation of the form y=mx+c from the above expression

[orchid mirage]

Oh I think I got it

[near atlas]

great

[vestal carbon]

Do you agree that for any point (x,y) on the line, the slop e of this point with origin will be same as slope of the given point with origin

[orchid mirage]

Is it something like this

[vestal carbon]

Yes

[orchid mirage]

slope of a straight line are equal for points on the line which are collinear no?

[vestal carbon]

Yes

Thats basically what youve done in your method

But to simplify it, you can find slope of (x,y) with origin

Instead of -2root2, -3root3

[orchid mirage]

I.e. here the slope is 3 √3 / 2 √2

if y=mx+c form

[vestal carbon]

Im saying that, your x1,y1 can be 0,0 as well

[orchid mirage]

oh wait if c=0 it passes through straight line

[orchid mirage]

yep it doesn’t quite matter

[vestal carbon]

It doesnt i know

Im just saying that will sinplify your calculations

[orchid mirage]

how can I solve F

[vagrant ice]

do you know the change-of-base rule?

[orchid mirage]

I don’t

what is it

[vagrant ice]

go and watch a video on it then

$\log_a (b) = \frac{\log_{10} b}{\log_{10} a}$

you don't need to use 10 for the base actually, can be any common base

[flat frigate]

south

[orchid mirage]

ah like

how do I apply it here

[vagrant ice]

so the first term would be ${\log (\sqrt q)^2} \over {\log \sqrt{p}}$

[flat frigate]

south

[orchid mirage]

oh I get it

answer is 8 I think

thanks

.close

[safe radish]

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[orchid mirage]

what is the slope of a line representing x-axis

is it 0?

[safe radish]

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[orchid mirage]

Seems very hard

[limpid lodge]

which part??

[orchid mirage]

c and b

just guide me what I need to do

[west wasp]

I meed help with knowing the size of the text on a clothing

[limpid lodge]

why dont u find PQ,QR,RP first

[west wasp]

Asap

[kind seal]

!occupied

[safe radish]

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).

[kind seal]

ch3rry is right, just find the products and you are almost done

[orchid mirage]

I don’t actually understand

yk

[kind seal]

$a^ma^n=a^{m+n}$

[orchid mirage]

like the way the equations are given

[flat frigate]

Adhi

[kind seal]

do you know this

[orchid mirage]

to derive the points

[orchid mirage]

I meant c

I can do b actually

[kind seal]

you done with b?

okay

[orchid mirage]

yeah

[kind seal]

let me read c

[orchid mirage]

Idk how to get the coordinates with them

[kind seal]

I assume P is the intersection of the lines

[limpid lodge]

have u drawn a diagram for c?

[kind seal]

Have you studied the area of a triangle using coordinates

[orchid mirage]

Yeah

It’s like 1/2

[kind seal]

Try drawing the figure and use trapezoids

[orchid mirage]

Then one of the repeatation

[kind seal]

yes

you can use that directly

[orchid mirage]

I need to know the coordinates

[orchid mirage]

I can’t because I don’t know how to get the coordinates

[kind seal]

I think, though not stated that P is the intersection of the lines

[limpid lodge]

hows P related?maybe they mean O?

[orchid mirage]

P is the origin

[kind seal]

ooh

so you know P

[orchid mirage]

yeah

[kind seal]

A is the x-intercepts of one of the lines

i.e. where they cut the line y=0

[limpid lodge]

have u solved ii?

[limpid lodge]

i mean drawn the diagram for it

[orchid mirage]

wait if one line cuts it at x then y=0 other line must cut it at y axis

So x=0

[kind seal]

yes

B is the y-intercept of the second line

[orchid mirage]

intersect x axis means y=0

[kind seal]

yes

[orchid mirage]

OH SO LIKE I PUT the value and solve the equation

?!?

[kind seal]

yes

[orchid mirage]

now I can

[kind seal]

(x, 0) with y=0 for the first equation is the first point

you can take it from here on ig

[orchid mirage]

wait it’s actually not that difficult

cherry do you need the channel?

[limpid lodge]

*didn't realize u were confused with the diagram *

[limpid lodge]

no

[orchid mirage]

oh

ok then thanks

thanks to you two

.close

[safe radish]

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[echo phoenix]

im on b
need help
idk what im doing anymore

[echo phoenix]

huh

[kind seal]

one sec

yeah ok, have you studied similar triangles

[echo phoenix]

yeah

[kind seal]

you know every vector in triangle MNQ (using the polygon law of vector addition), you also know vector NR

[echo phoenix]

mhm

OH

OKAY THANK U

b/4? @kind seal

[kind seal]

@echo phoenix the question wants OS

you are almost there

you dinf RS correctly

[echo phoenix]

oh so RS+OR

[kind seal]

bingo

[echo phoenix]

so it would be 5b/4 right

[kind seal]

correct

[echo phoenix]

okay thanks

.close

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[thin lion]

Hey, I was wondering do I need a sum sign on the 1/2n -1

It doesn’t feel right

[safe radish]

@thin lion Has your question been resolved?

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[limpid lodge]

How do u simplify this without putting in the value of omega

[limpid lodge]

[
(1+\omega)^n + (1+\omega^2)^n
]

[flat frigate]

ch3rry

[winged flare]

$\omega ^2 + \omega + 1 = 0?$

[limpid lodge]

Yes.

[winged flare]

$\omega + 1 = -\omega ^2$

[limpid lodge]

Mmhm

[winged flare]

yeah proceed

[limpid lodge]

Proceed TT

(-w²)^n + (-w)^n

[winged flare]

cool

take (-1)^n common

[limpid lodge]

w^2n + w^n

[winged flare]

yeah it depends on the value of n from here

you could explain with cases

[limpid lodge]

Tht is what i dont understand

[winged flare]

if n is a 3 multiple it would be $(-1)^n(1+1)$

[limpid lodge]

I put in the value of omega and got 2cos(npi/3)

[limpid lodge]

Okay

[winged flare]

oh waittt

[limpid lodge]

How do u get to this now?

[winged flare]

cant recall how to get this

but the cases would match

[limpid lodge]

[winged flare]

guessing you use $-\omega = -(\cos(\frac{2\pi}{3})+i\sin(\frac{2\pi}{3}))$

[winged flare]

and then de moivres

the sin terms cancel

and you get that

this is de moivres

and modulus of omega is 1

so r = 1

[limpid lodge]

Well thts putting in the value of omega

[limpid lodge]

I wanted without the value

[winged flare]

not sure if possible

feels like question intended to use de moivres

with the 2npi/3

[limpid lodge]

I did get the answer by using tht. But i js wanted a method without plugging in value.

<@&286206848099549185>
...?

[dark isle]

Yes

[kind seal]

If you want the answer as a cosine you have to put the value

[limpid lodge]

In what other terms can we get the answer?

Brb

[kind seal]

I think this is maybe the only way

[safe radish]

@limpid lodge Has your question been resolved?

[limpid lodge]

Sure??

[drowsy slate]

hi

whats ur question?

\

[limpid lodge]

Its pinned

[drowsy slate]

ok wait

oh this is not my lvl

srry

[limpid lodge]

[drowsy slate]

im srry

im in 9th

i can help more simple

[limpid lodge]

Its alr

[primal bone]

Surely from this you could continue geometrically?

[limpid lodge]

Ehh im awful with geometry would u elaborate??

[primal bone]

As in on an Argand diagram

oh wait hang on lemme check what this omega should refer to

Cube root of unities, right?

[limpid lodge]

Yes

[primal bone]

You know where those are on an Argand diagram?

[limpid lodge]

Lemme look at my notes

Erm all ik tht when plotted they represent vertices of equilateral triangle

[primal bone]

ye

Specifically they're on the unit circle

Do you recall what happens with the mod and arg of a complex number if we raise it to a power?

[limpid lodge]

No

[limpid lodge]

Yes

[primal bone]

So what would happen if z = w

[limpid lodge]

Doesn't argument have similiar properties with logs?

[primal bone]

I mean, in so far as "log(xy) = log(x) + log(y)", yes, but that's what I wrote, isn't it

[limpid lodge]

Yes

So 2arg(z)??

[primal bone]

ye

And for the mod?

[limpid lodge]

Modz² ?

[primal bone]

(the | should be where your \ key is)

but ye

[limpid lodge]

|z|²

[primal bone]

ye

[limpid lodge]

And rhs |z²|

[primal bone]

Now, we're dealing with stuff on the unit circle, so the mods are all 1 anyways

That makes our job easier, because we only have to care about the args

[limpid lodge]

Okay well. Where r we going with this?

[primal bone]

We want to justify where the omegas land when we raise them to powers

(actually, lemme reflect the diagram a bit, since we have (minus omega) and (minus (omegasquared)))

[limpid lodge]

( )

[primal bone]

there we go

[limpid lodge]

should i js continue with other topics?

[primal bone]

I mean, technically you have an answer here anyways

[limpid lodge]

Where😭

[primal bone]

What I responded to

[limpid lodge]

Hows tht the answer?

[nimble vine]

Sup, btw we still on that?

[primal bone]

That's what I'm trying to explain

[limpid lodge]

[primal bone]

Not conductive to productive discussion

[primal bone]

You know how we have (-w) and (-w^2), and we're just raising these to powers and adding them? @limpid lodge

[limpid lodge]

Yes

[primal bone]

It helps then to check where those powers send those numbers, and then what the addition does

We've already noted that the mods aren't gonna change, because it's gonna stay on the unit circle

(er, technically not, we've only checked squares, but this generalises quite neatly, so:)

[limpid lodge]

Ok

[primal bone]

Can you check the args of -w and -w^2?

[limpid lodge]

If we make cases putting n=3k,3k+1,3k+2

Where k is an integer

[primal bone]

ye...

[nimble vine]

Wdym

[limpid lodge]

Then.....TT

[primal bone]

"We're still on this?" is just an assertion that the helpee is taking a long time, which doesn't help them

[primal bone]

Well, as you've said, cases

Let's start with n = 3k

[limpid lodge]

Yeah well get 3values

[nimble vine]

Oh mb i was asking we talking on that, or another q

[limpid lodge]

But we also have (-1)^n TT

[primal bone]

Focus on the fact that (-w) is just a complex number we're raising to a power

If it helps, let z = -w

[limpid lodge]

This wouldn't work ryt

[primal bone]

yeah, I'm thinking it's closer to 6 cases

(which I think actually collapse down to 2, but still)

Here's a diagram exploring the powers of (-w); idk if it makes any sense

[astral glacier]

What question are we on rn

The pinned one still?

[primal bone]

We're simplifying this, ...

[astral glacier]

Ah cool the pinned one

[primal bone]

...and it's been reduced to this

oh shit wait, yh we could just pull out a (-1)^n

[astral glacier]

Yes

[vagrant ice]

you guys are still working on this

[astral glacier]

Not just that, you can invoke the factorization of a^n + b^n

[primal bone]

oh ffs

[astral glacier]

And then the first term becomes -1 again

And the second term is "cyclic" in nature

So your eventual solution is smth to do with n mod 3

[vagrant ice]

I mean the key is to realise that $(\omega^2)^2 = \omega \cdot \omega^3 = \omega$

[flat frigate]

south

[astral glacier]

w^k = w ^(k%3) is more like it

[vagrant ice]

so you get this yes

[primal bone]

@limpid lodge I've written out, carefully, how the (-1)^n can get pulled out; can you see how to continue this?

[tho tbh I'm also getting a little lost ]

[limpid lodge]

No this i understand

U didn't need to write it carefully TT

[primal bone]

idk why, the other ppl talking about the length of time is peeving me, so

[limpid lodge]

Eh its not been tht long tbh

[primal bone]

The pin is from 1h36m ago

I suppose for some that is long for this

[limpid lodge]

Haven't been active every second of tht

[primal bone]

But yh acc with the factorising it just boils down to making sense of w^n + w^2n

[limpid lodge]

But thts not thr pointtt

[astral glacier]

Don't sorry about that lmfao

[astral glacier]

What exactly are you confused about then

[limpid lodge]

Can someone just provide me th soln 🤧

[astral glacier]

!noans smh

[safe radish]

The purpose of this server is to help you learn, not to hand out answers. Do not ask someone to give you the answer directly.

[limpid lodge]

I already got my ans. I want to try a diff method.

[astral glacier]

Well we have told you quite a bit about a different method

[primal bone]

@limpid lodge Here's a more useful diagram, based on this being the main hurdle

[astral glacier]

Factorize and then use the cyclic method

[limpid lodge]

Ohk.....

Just provide me the full soln. Then ill ask where i dont understand??

[primal bone]

Xavier you are one step away from "just use Sridharacharya"

[primal bone]

Can you see that if the n-power is a multiple of 3, that the addition of the two things is pretty clear?

[astral glacier]

Unfortunately as per server policy, we aren't allowed to hand out solutions to helpees

!nosols for reference

[safe radish]

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.

[limpid lodge]

Well its clear for all of these

[primal bone]

If it isn't a multiple, than precisely one of them is a w, and the other is a w^2

[limpid lodge]

True

[primal bone]

So their sum is already mentioned in w^2 + w + 1 = 0

[limpid lodge]

Mmhm

In what terms will we get our ans using this?

[primal bone]

So what would their sum be if a) n is a multiple of 3, b) n is not a multiple of 3?

[limpid lodge]

Ignoring the (-1)^n?

[primal bone]

ye

we'll bring that in later if we have to

[limpid lodge]

Oh

Wait i gtg again.

Ykw ill js skip this

[primal bone]

If you open a new channel coming back to this, try and restate what you can in the new channel (this channel should still be read-accessible to you, if it helps )

[limpid lodge]

Eh maybe ill be bk in an hour or so

Im gon keep it open

[nimble vine]

as you wish

[safe radish]

@limpid lodge Has your question been resolved?

[rigid blaze]

bump

[limpid lodge]

?

[rigid blaze]

it would get closed by timeout or smth or so I thought, so I decided to just say bump to keep the thread alive

[normal moss]

They reacted with an ❌

It would have gotten closed had they not responded

[rigid blaze]

oh

[limpid lodge]

..well im bk

[primal bone]

^

[limpid lodge]

?

@primal bone

[primal bone]

Can you work out what the expression then equals, for n = 1, 2, ..., 6, then?

[limpid lodge]

So like what now

How would tht gelp

Help

[primal bone]

If n = 1, is that odd or even?

[limpid lodge]

Odd

[primal bone]

So what does (-1)^n equal?

[limpid lodge]

Well for odd n the expression is multiplied by -1 and opp for even

So now how does tht helpppp

[primal bone]

wdym "how does that help"

[limpid lodge]

As in are we getting closer to our answer?

[primal bone]

answer this first

[limpid lodge]

For odd? -1

[primal bone]

And to which of these cases does n = 1 adhere?

i.e. is it a 3k, a 3k+1, or a 3k+2

[limpid lodge]

2nd

K=0

[primal bone]

ye

So what does w^2n + w^n evaluate to?

[limpid lodge]

1

[limpid lodge]

Uhh yes. Tht into(-1)^1

[primal bone]

The whole thing is -1x-1 = 1, yes

But RTFQ

You keep jumping all around the question

[limpid lodge]

What ques TT

[limpid lodge]

This?

[primal bone]

This is a question, we're trying to get through a solution, and you are jumping all over trying to do this thing then that thing, without seeming to be paying attention

In any case, we've got that the whole expression is 1 for when n =1

[limpid lodge]

I dont understand what you're trying to do

[primal bone]

In much the same way, ^^^^

The WHOLE POINT of this problem is to figure out what this thing equals, right?

[limpid lodge]

Yes....simplify it

[primal bone]

!xy

[safe radish]

Please show the original problem, exactly as it was stated to you, with the entire original context. A picture or screenshot is best. If the original problem is not in English, then post it anyway! The additional context might still be helpful. Do your best to provide a translation.

[limpid lodge]

Oh its a part of binomial series

Idt i hv the whole ques rn.

[primal bone]

...

With all the due respect I could muster, this would have helped a lot if you a) had this question and b) said this not 4 hours ago

This can be simplified to the point of just evaluation, which is precisely what we're doing

[limpid lodge]

Idts but alr.

[primal bone]

To the same damn extent, ^^^^^^^

[limpid lodge]

Idt we're getting anywhere, i got my ans by plugging in the value of omega

So ima stick to tht thnx anyways

.close

[safe radish]

Channel closed

Closed by @limpid lodge

Use .reopen if this was a mistake.

[nimble vine]

eh cherry you there

[limpid lodge]

Yes?

[nimble vine]

$\omega^{3n+k}=\omega^k$

[flat frigate]

oppenheimer

[nimble vine]

did you try this?

[limpid lodge]

...yes i believe

When making cases

[nimble vine]

so.. it should work

[limpid lodge]

..we made the cases . Are those cases the answer?

[nimble vine]

you deduce from those

eg w^4 = w

[limpid lodge]

Wdym

[nimble vine]

in w^2n + w^n put n=4

which is 3k +1

w^8 + w^4 => w^2 +w

which is -1

[limpid lodge]

Ok

Bruh cant believe chatgpt explained better than all ppl here. I actually got the other method.

[nimble vine]

you understood is what matters

[limpid lodge]

Oh yes. I regret sm not asking chatgpt in the first place i swear, everyone here made me go haywire for no rsn lmao.

[nimble vine]

lmao ppl here try to help instead of directly explaining the soln, so you get it yourself

[desert pasture]

!nogpt

[safe radish]

Please do not trust ChatGPT or similar AI tools for mathematical tasks, as they often generate output which "sounds correct" but has numerous factual or logical errors. Use of these AI tools to answer other people's help questions is strictly against server rules (see #rules).

[desert pasture]

that's how you learn

[limpid lodge]

Maybe the their "teaching style" didn't mathc my "learning style"

[winged flare]

the ragebait is crazy

[nimble vine]

chatgpt mockin us

[winged flare]

what'd chatgpt do 💔

[nimble vine]

exist

[limpid lodge]

I didn't mean it tht way

[winged flare]

yeah i get it

[twilit wind]

ok you got it

nevermind

[safe radish]

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[covert yoke]

If we have any homomorphism between groups $\varphi: G \to H$, and we consider the fibers of $\varphi$ in $G$, is it true that the cardinality of each fiber in $G$ is either 0 or $\alpha$ for some fixed cardinal $\alpha$?

[flat frigate]

OmnipotentEntity

[covert yoke]

My gut said only alpha, but then I realized that non-surjective homomorphisms exist, so I had to modify it to 0 or alpha, and now I'm not 100% convinced. I attempted to show that it was by considering the group structure of H, but I don't think that the group action of h on the fibers necessarily must be bijective between fibers.

(This isn't an exercise, just an attempt to reason through a definition, and see if it behaves the way I think it ought to behave.)

[peak estuary]

all fibers are cosets of the kernel

[hardy lion]

whats a fiber with cardinality 0 look like?

oh right

nvm

[hardy lion]

this

[covert yoke]

If phi is not surjective then there exists h in H such that phi^-1(h) = {}

[hardy lion]

you can show by first iso thm that fibers are cosets of the kernel

[covert yoke]

Thanks!

I got to the end with this.

.close

[safe radish]

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[exotic charm]

2b pls

[exotic charm]

ive tried for a very llong time but im getting nowhere

[safe radish]

@exotic charm Has your question been resolved?

[still sentinel]

What happens if $$E \cup N \in R$$? What property of N changes? (Or what are we allowed to do now with N)

[flat frigate]

DoctorFuchs

[exotic charm]

uhh then (E U N) \ E = N\E in R ??

[still sentinel]

Wait I misread it, sorry. Let me think about it. Give me 2 minutes.

[cold relic]

Can we write E = E\F u EnF ?

Then distribute M (subset of F)

[still sentinel]

That could work, I wondered if it would be helpful to make a distinction into different cases, but going with F is smarter ig.

So E\F is in R. So we have to show that (E n F) u M is a null set right?

[cold relic]

\M right?

Since EnF has measure 0 (EnF)\M should be a null set

[still sentinel]

True sry

Yeah that works. Do you understand it @exotic charm

[safe radish]

@exotic charm Has your question been resolved?

[exotic charm]

lemme see

[exotic charm]

not rly

[exotic charm]

i acc managed to show it my own way tho

[safe radish]

@exotic charm Has your question been resolved?

[wet wigeon]

do you need anyone to verify or are you done?

[still vine]

and if you are done please close ticket for other people

[wet wigeon]

^
@exotic charm

[long oracle]

Did u want to see my solution to this btw

I know we talked about this yesterday

Or how did u show it?

[safe radish]

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[toxic sage]

Can smbdy help me w this (I’m stupid af)

[quasi bison]

do you know what "parallelogram" means

[toxic sage]

Yes, I can’t find on yt how to do ts typa thing

There’s onky 2 givens in which I could find

[quasi bison]

don't call me "bro" please.

[toxic sage]

Mb

[quasi bison]

could you edit that word out please

can you tell me what the word "parallelogram" means though

or some properties of parallelograms

do not worry about applying them to this question yet

[toxic sage]

All i know is they’re a quadrilateral

[quasi bison]

there are many types of quadrilateral

can you tell me what the requirements are to be called a parallelogram specifically? yes or no

[toxic sage]

All i know is that their adjacent angles have to be sum up to 180 degrees ig

[quasi bison]

then your previous statement, "all i know..." was not true. it wasn't all.

[quasi bison]

that is correct and allows you to find n.

adjacent angles add up to 180°, in particular angle M and angle E

@toxic sage would you like to find n first, or talk about the sides first?

gonna need to do both but you're deciding the order

[toxic sage]

Let’s go with n

[quasi bison]

ok

[quasi bison]

can you write this as an equation

don't solve yet

[toxic sage]

Wait

Did i write it right

excuse the bad handwriting

[safe radish]

@toxic sage Has your question been resolved?

[blissful tapir]

Why the extra m?

[safe radish]

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[safe radish]

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[oak scroll]

Can someone help with these questions, I am confused and don't know where to start.

[cunning knoll]

for the second one substitute t=1-x

[oak scroll]

oh so the whole thing but inside the brackets, its just t?

[cunning knoll]

no i meant

if u substitute t=1-x

and diffrentiate it

and write the integral in terms of t

u will find ur ans

[quasi timber]

The integral is non elementary. No substitution or trick is going to work.

[cunning knoll]

if u substitute it u will get -I

I=-I

I=0

[quasi timber]

that’s not correct.

[cunning knoll]

wait ill share a pic

[oak scroll]

∫01​lnxln(1−x)dx=2−6π2​​

i got something like this

[quasi timber]

I don’t think you understand — the issue is that no trick or substitution or whatever will work unless you use the polylogarithm, which is a non elementary function. I do not need a picture.

[quasi timber]

you mean

$∫_0^1 \ln(x)\ln(1-x)\di x=2-\frac{π^2}{6}$

[flat frigate]

𝙸𝚗𝚏𝚒𝚗𝚒𝚞𝚖³

[quasi timber]

?

[oak scroll]

yesss

yeah

[quasi timber]

that is correct yes.

[oak scroll]

its like a weird decimal

0.35

onwards

how do I do the other one

[cunning knoll]

Oh mb it's symmetric under substitution

[ornate glacier]

Represent the partial sum as an element of a sequence. Use the stolz-cesaro theorem after.

[oak scroll]

alright thanks I got the limit is 2

.close

[safe radish]

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[safe radish]

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[glacial plume]

a bird is resting on top of a skyscraper that is 108ft tall. he flies 51, ft to the top of the next building which is 64 ft tall. how far apart are the two buildings

[glacial plume]

I am low-key confused asf

[novel lichen]

yo bro i need help

do yalll know about comparing quantities

[safe radish]

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).

[lost jewel]

Make a diagram from what you undertand from the problem

If it's wrong, we'll help you correct it

[glacial plume]

I think?

I am low-key stupid asf

.close

[safe radish]

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[safe radish]

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[fierce ravine]

someone help please

[stiff magnet]

does riemannian technique work/

?

[cold ridge]

i think you can do feynman's technique on this

[visual linden]

Not familiar with this, but first thing I'd do is take ln(x) = u

[stiff magnet]

yah

[cold ridge]

you have a nice parameterisation with that e^(-ax^2)

@fierce ravine are you familiar with feynman's technique of integration?

[stiff magnet]

https://www.youtube.com/watch?v=YO38MCdj-GM @fierce ravine if u r not

[vestal carbon]

That wouldnt help with anything

[visual linden]

Alright, the methods suggested by the other two guys then

[cold ridge]

^^ but this is also an amazing video that i recommend, it was how i learned feynman's trick https://www.youtube.com/watch?v=wOn9r-YWj3Y

[stiff magnet]

an no prob

my inclination towards ma asian brather

[cold ridge]

still the same tho, i just think bprp goes straight into it without any explanations for beginners who aren't familiar with it so far

[safe radish]

@fierce ravine Has your question been resolved?

[safe radish]

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[lapis token]

How do I prove the determinant gets multiplied by -1 when you swap two rows OR columns in a matrix

[lapis token]

Can it be done by applying the laplace expansion on one of the columns?

[peak estuary]

which definition of the det do you have

[lapis token]

Geometrically?

Well either way our definition looks like this:

There is no explanation for it

It’s just the formula and it says it’s called the determinant

I know what it geometrically refers to but I’m not allowed to use that because it’s not a part of the script as far as I know

[peak estuary]

ok so leibnitz formula

[vestal carbon]

Lol i keep forgetting we have to prove for n by n and not just 3 by 3

[lapis token]

Same tho
I was going to just use saruss on a 3x3 and pretend to have a happy end

[peak estuary]

consider pi(tau(i)),i where tau is the transposition of the rows/columns

or tau(pi(i)),i

I always forget the order

[lapis token]

Before we keep going

We have the laplace expansion available too

And I’m a lot better with that than I am with this formula

Can I use that instead?

[peak estuary]

that should also work

expand along any row/col you want

use induction on the smaller dets that appear

keep track of all the signs

[lapis token]

Wdym by this

[peak estuary]

induction over n

when you do laplace you get dets of size n-1

so by induction hypothesis you can assume the claim is true for those

[lapis token]

So I prove that it’s true for a 2x2

Say you can technically just split a 3x3 into 9 different 2x2

And a 4x4 into 16 3x3

And so on?

[peak estuary]

laplace says that you can compute a 4x4 det using 4 3x3 dets

[lapis token]

So it’s ultimately enough if I say

2x2 satisfies the condition (proof)

So 3x3 satisfies it

Because it can be computed using 2x2

Which means that 4x4 […]

So an NxN is also satisfying the condition

[peak estuary]

you still have to show that the details work out

[lapis token]

For the 2x2 right

[peak estuary]

for every step

[lapis token]

And then say why this proves 3x3

[zinc hornet]

If you want to proceed with this strategy, you really need to first learn what induction is. The threshold for what is enough is when you apply induction correctly.

[lapis token]

I know induction as:

Show n=1

Show why n=1 ==> n=2

Show n=x ==> n=x+1

Ignore the notation

I know it’s messed up

[verbal cloud]

You usually don't need this

[verbal cloud]

It's covered in this

[lapis token]

I might misremember then
That’s convenient tho

[safe radish]

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[clear helm]

wat?

[novel root]

What

[fathom jewel]

Whats ur question

[clear helm]

how to solve it give your example

i'm not sure

[scarlet plank]

Identify the type of recurrence. Try solving the homogeneous part first

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@clear helm Has your question been resolved?

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[trail salmon]

Can someone help me understand what this is asking?

[trail salmon]

I think I found the answer to how to do this in my notes

is this right?

<@&286206848099549185>

I might have drawn a few too many normals but

[safe radish]

@trail salmon Has your question been resolved?

[safe radish]

@trail salmon Has your question been resolved?

[last wren]

yes that is correct. some of your normal vectors look a little wacky but i'm sure that's fine lol

[trail salmon]

yeah drawing on mouse is a little wonky

okay cool glad I understood it

[trail salmon]

.close

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[quiet jewel]

For all 5 I have to determine if it's exponential, quadratic, or linear and I'm pretty sure I got like 4 wrong

[quiet jewel]

Like I get A since yeah, but B-E I don't get

Specifically D, why is something cooking not linear

Is it because it approaches something but never gets colder than the diner?

[autumn sun]

well, have you learnt about the very basics of thermodynamics?

if not, consider how fast a piping hot drink cools in the open, vs. how fast a warm drink cools

[viscid jay]

D and E definitely feel like weird questions for a pre calc class

For B think about how increasing the number of teams by one would change how matches a team would play

[quiet jewel]

Uhhh

Is it that like

There are 2 more games

[viscid jay]

Yeah!

[quiet jewel]

Okay cool

But then C

The total games

I thought it was exponential

[viscid jay]

Do you know any combinatorics? There’s a very handy equation for this

This is showing the amount of unique ways to pick k amount of things from a pile of n things.

So I’m our case k should be two

and n should be the number of teams

That’s the number of matches we get by each team playing every other team once

Try to plug-in in k and simplify

[quiet jewel]

uhm

[quiet jewel]

we never learned any of that

i think we will learn it next semester

well probability i think thats what im seeing

[viscid jay]

Imagine a grid where all the teams get one row and one column. Each square signifies a match between the team that has the the squares row and the team that has the squares column, excluding the squares that have the same team in a row and column.

[quiet jewel]

uhm

can i just thijnk it doubles every time

[solar hazel]

[quiet jewel]

like 2 teams will play 4 matches right?

[viscid jay]

Imagine adding more teams

Sorry I think I’m explaining this badly lol

[solar hazel]

you don’t need any choose function for any of these

[quiet jewel]

what?

[quiet jewel]

okay wait i kinda see it

bc 2x2 grid will make it 4, then 9, then 16

what about that slingshot one?

[quiet jewel]

i was thinking it would be linear right? like i pull back with 1N then it will go back at 1N but idk

[autumn sun]

are you meant to look up the formula for elastic potential energy for that?

[quiet jewel]

i dont think so

[autumn sun]

then this is kinda weird

[quiet jewel]

im given 5 formulas after i do all 5 situations

ill take a pic

bc im confused on this also

[autumn sun]

,rccw

[autumn sun]

if you can do the first part you should be able to do the second

[autumn sun]

but I will tell you this

EPE (elastic potential energy) is not a linear relationship

[quiet jewel]

what is it? our teacher has never even talked about it

[autumn sun]

the EPE of an object increases with the square of the extension of the object

[quiet jewel]

oh so quadratic

okay i have the 5, i have this order left to right

D? A _ E_

idk the 3rd or 5th

is 5th C?

just bc its quadratic

[autumn sun]

try testing with some small values

[quiet jewel]

2 teams, 2^2 -2

[autumn sun]

for C, I will tell you that two teams create two games and 3 teams create 6 games

4 teams will create 12 games

if you can verify these against equation 5, then yes, C is the right fit

[quiet jewel]

why does it have -x and -2

[autumn sun]

where

equation 5 is x^2 - x

[quiet jewel]

like on the formulas

yeah, why -x

[autumn sun]

it's kinda hard to show this without combinatorics

but either way, if it works, it fits

[quiet jewel]

2 teams, each play against each other 2 times would be 4 then - 2 = 2

[autumn sun]

and indeed 2 teams should play against each other 2 times

[quiet jewel]

wait im slow yeah

3^2 -3 = 6

[autumn sun]

nono, it's ok, I'm just confirming it for you

[quiet jewel]

im trying to understand it hold on

[autumn sun]

I'll put a check if you're right

[quiet jewel]

how is it 3 teams play 6 games?

lemme draw it or something

[autumn sun]

A plays B, A plays C, B plays C

so 3 matchups

but each matchup plays 2 games

so 3x2 = 6

[quiet jewel]

okay i see it

man thats weird

[autumn sun]

combinatorics work in weird ways

that's why counting is hard

but it's not impossible

[quiet jewel]

i get 2 since yeah it just doubles, 3 teams, one team will play 4 games

[quiet jewel]

situation 2*

okay yay i think i can do it now

thank you for all your help

have a good night

.close