help-36

[copper roost]

so look

i have abitur

[supple jolt]

yes

[copper roost]

in 6 days

[supple jolt]

haha

[copper roost]

and i dont know alot

i didnt even start vectors

[supple jolt]

its ok

[supple jolt]

ah shit

[copper roost]

so like

is it too late

or

[supple jolt]

hm

pff

was soll ich sagen

welches von den 3 wird mehr bewertet

analysis, geometrie oder stochastik

geometrie oder

[copper roost]

boah keine ahnung

[supple jolt]

ich glaub schon

egal

em

ich empfehle dir auch etwas geometrie zu lernen

[copper roost]

warte kannst du das nochmal erklären mit f(2)

[supple jolt]

denn sonst

[copper roost]

100%

mach ich noch

[supple jolt]

ja also

[drowsy epoch]

ja erklär mal

[supple jolt]

star are you also german

[drowsy epoch]

ja

[supple jolt]

crazy

also f ist die ableitung der gesuchten funktion

f zeigt die steigung von F

ok bis jetzt?

[copper roost]

ja

[supple jolt]

gut

also wenn wir f(2) zb berechnen

[copper roost]

ja

[supple jolt]

entspricht das der steigung von F

an der stelle 2

[copper roost]

ahh okay

[supple jolt]

ok berechne mal f(2)

[copper roost]

-3

[supple jolt]

1/4 * 8 - 6

2-6

-4

[copper roost]

nein ich mene

[supple jolt]

oder

[copper roost]

sorry

ja

genau

habs im kopf gemacht

[supple jolt]

k

also F sollte die steigung -4 an der stelle 2 haben

guck mal auf den graphen

welcher passt

[copper roost]

wie soll man das aber ablesen??

ich würd graph 2 schätzen oder

[supple jolt]

<@&268886789983436800>

[supple jolt]

genau

-4 heisst ziemlich grosse steigung

steiler graphen

[copper roost]

ahhh

und weniger wäre dsnn

breiter oder wie

?

[supple jolt]

weniger steil

vielleicht so was

[copper roost]

aber die steigung kann man nicht ablesen direkt oder?

[supple jolt]

kannst sehen das rot weniger steil ist als II

[supple jolt]

doch

[copper roost]

doch ja

eigentlich schon

ja hast recht

[supple jolt]

delta y / delta x

und wir haben -4

also ist die steigung negativ

[copper roost]

ja klar

[supple jolt]

zb bei x=4 ist die steigugn positiv

[copper roost]

könnte man die funktion integrierenn

[supple jolt]

klar

[copper roost]

und dann schauen was vor x^4 steht

[supple jolt]

versuch's einfach

[copper roost]

ja mache es gleich

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[mild prism]

How do i use Π with the line on the top longer

[night raft]

this? $\prod$

[soft zealot]

MxRgD

[mild prism]

Ye

[night raft]

you can copy and paste capital pi

actually

Π yeah it just does the same :/

[mild prism]

Ok how does it work

[rare girder]

the product symbol is just a capital Pi anyway

P, for product

[severe canyon]

Do you know $\sum$ symbol?

[soft zealot]

Alberto Z.

[mild prism]

Σ

Yes

[sterile marsh]

instead of adding just multiply. that's all

[mild prism]

Ohh ok

Also whats a log in math

[severe canyon]

It's the abbreviation of logarithm, which is the inverse function of an exponential

[sterile marsh]

a^b = c
so, log(a)(c) = b
basically, a should be raised to the power of b to obtain c

[mild prism]

Oh oke

[sterile marsh]

!done

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[mild prism]

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[onyx peak]

<@&268886789983436800>

[chilly granite]

<@&268886789983436800>

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[opaque ember]

lol

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[vital crag]

<@&268886789983436800>

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[crystal lion]

how and where does this question allude to using methods of differences to find part a

[rugged merlin]

Partial fraction decomposition

Realise that the denominator of that fraction can be factored

[crystal lion]

how would u know to do that?

[rugged merlin]

Which can then be decomposed

Which the results in a telescoping sum

[rugged merlin]

Thats like number 1 thing you check with sums

Check for a telescoping sum

[crystal lion]

ohhh

okok my brain immediately went to the standard form part of summations like just somehow using the identities for r = 1/2n (n+1) etc, but i couldnt see how u would be using it for this case

thankss!

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[fresh pendant]

How do I get the intercept and slope from summary(linear regressional model)

[vital crag]

Show the context you have

[fresh pendant]

I'm doing question 3 atm

[fresh pendant]

Lemme know if u can help or not

I'm guessing B Bar naught is 1.708e+2

And that B Bar one is 5.232e-03

Therfore 1.708e+2 + 5.232e-03 x 174mm

The answer would be 171.7104

Or rounded 172

@vital crag mate

What do u think

<@&286206848099549185>

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[fresh pendant]

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[dense badge]

this seems good if you want more precision there is a method for that in R : predict(model_object, values_to_predict)

[fresh pendant]

Alrighty thanks

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[unkempt prism]

I want to calculate the work done $\int_S \vec F(\vec r) \dd \vec r$ with $\vec F = (x + z, z, x + y)^\top$ and $\vec g(t) = (3, 0, 6t)^\top$ along the striaght path from $(3, 0, 0)$ to $(3, 0, 6)$. Does $$\int_0^1 \sqrt{(3 + 6t)^2 + 36t^2 + 9} \cdot 6 \dd t$$ look fine?

[soft zealot]

ILikeMathematics

[final saddle]

@unkempt prism Has your question been resolved?

[drowsy epoch]

\prpl I dont think you are supposed to take the magnitude, the formula was
[ int_{C} mathbf{f} dotproduct dd{mathbf{s}} = int_{alpha}^{beta} mathbf{f}(mathbf{g}(t)) dotproduct dot{mathbf g}(t) dd{t}. ]

[unkempt prism]

Yes but then what is f(g(t))?

f is a vector

Then I would get a vector as work done

[drowsy epoch]

dr is also a vec

[unkempt prism]

we dont have dr anymore though

on the RHS of this

[glossy zephyr]

g dot t also is

[unkempt prism]

oh

[glossy zephyr]

more specifically, they both create a dot product

So you end up with a scalar magnitude

aka, work

[unkempt prism]

Thanks!

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[warm python]

Here I suppose I'd want the two sample t-test?

[warm python]

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[warm python]

.reopen

[final saddle]

✅ Original question: #help-36 message

[warm python]

So here if I can't assume equal population variances, but given the small population size what test do I use

the Z tests require sample sizs of atleast 30

[nimble shuttle]

dont you do x-y and find the pooled mean and variance

To get the distribution you want to work with

[warm python]

hmm? but here the groups are indpendent

so it's not a pooled test

oh pooled

not paired

[warm python]

I don't get this though, for pooled variance we assume equal pop variances

[nimble shuttle]

oh really

idk then ig

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[late gazelle]

If an ant walks from the base of a right cone to the other side of the cone, what is the shortest "walk" the ant makes. Given that the base of the cone is pi units long and it takes a total of 4 units to walk up the cone and back down?

I am sorry, I do not have the exact question/wording. The shortest "walk" i've found is a parabolic shape which is slanted on the cone but im not sure how to find the length of said parabola (if that is the correct method)

[thin cloud]

other side of the cone?

[late gazelle]

yea ill get a quick diagram

1sec

[chilly granite]

i think the path you found is correct-

[late gazelle]

yea but idk how to find the length of it

[thin cloud]

flatten the cone

[chilly granite]

-# yes but then the shortest path would be straight across which isnt what we want here..

[thin cloud]

huh?

[late gazelle]

but that straight line becomes a parabolic shape since the 3d surface is compressed into 2d - when extended back into 3d

[chilly granite]

yes..

[thin cloud]

[chilly granite]

-# woah nice i didnt think of this

[late gazelle]

yea make a net of the shape was one idea i had

well kind of

but same problem, idk how to find the distance

[thin cloud]

chord segment

[late gazelle]

well ik what it is but idk any relevant formulae

[thin cloud]

well first off, what's the length of the big chord

[late gazelle]

the big chord?

[thin cloud]

the red one here

[late gazelle]

ah yes that one

that should be pi

if i have understood the question correctly

[thin cloud]

yeah...

pi unit long

are they referring to the radius, diameter

or the perimeter?

[late gazelle]

i honestly dont know, the "base" is pi units long so i assume the circumference

since the ant cannot walk underneath the cone

or through it

[thin cloud]

okay

so what's the angle subtended by the chord

[late gazelle]

remind me what subtended means

[thin cloud]

blue one

or alpha

[late gazelle]

that should be 90deg

since its a "right cone"

right?

[thin cloud]

nah

actually the red one's called arc

[thin cloud]

there is a formula for length of a circular arc

You can find alpha using that formula

[late gazelle]

l = theta /2

i think

if l = pi

wait no

r theta = l

[thin cloud]

yup

r is l in my diagram btw

[late gazelle]

mhm

so depending on what the question means by 4 units to go up the cone and back down

so either the slant height l (in this case) is 2

or the distance from the base to the apex is 2

[thin cloud]

it's the former I believe

[late gazelle]

mhm that would make more sense

since ant cannot walk through the cone

so if r = 2 and l = pi, then theta = l/r

so pi/2

or 90deg

[thin cloud]

keep it pi/2, you don't have to convert it

[late gazelle]

typo*

so yea, a right triangle is at a 90deg angle

makes sense

[thin cloud]

so that mean angle subtended by the short path is half that

pi/4

[late gazelle]

yes

another word we could use would be bisected?

or is that in other contexts

[thin cloud]

well..., yeah? cuz start and finish are at the opp side so yeah

[late gazelle]

mhm ok

ok so we make a triangle thingy

we know one angle and one side

and it meets at 90deg to the base as well right?

is we draw a line from the apex to the base

[thin cloud]

we actually know 2

Sorry I'm having food atm

[late gazelle]

its allg

[late gazelle]

do tell (when your free ofc)

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@late gazelle Has your question been resolved?

[late gazelle]

.close

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[copper roost]

can someone explain when you write +C when integrating?

[drowsy epoch]

You basically always do

but when you have integration bounds, C cancels away, so we never write C in ther first place and just omit it

[supple pawn]

there could be a constant when you integrate yk?

[quiet python]

think about differentiating a constant, it gets removed, which means when integrating you don't know whether there was one or not

[copper roost]

wait so

[copper roost]

yes this makes sense

but how do u figure the constant out

if there was some normal number

like 5

bro im so stupid

[brazen breach]

we have definite integrals for that, indefinite have +C

[copper roost]

i forgot everything

[copper roost]

ohhh

right yeah

[night raft]

usually they give you a coordinate and you plug it in

that the equation passes through

[supple pawn]

indefinite integrals means to find what was your original equation

[brazen breach]

it's actually professor propaganda to make you lose points in the exam when you accidentally didn't write +C /j

[supple pawn]

yes

i hope u understand now

[copper roost]

yeah i gotta take a look at it again ill just do vectors and that stuff now

i mean i dont even know

how to study

[supple pawn]

alr you got worksheets on it?

[copper roost]

ngl no

only teachers notes

[supple pawn]

alr if u need help send it here

[copper roost]

like should i learn a thing till im perfect at it or do i just look at it a couple of times

[brazen breach]

ochem tutor and redbull gives you wings

[quiet python]

do a bunch of practice questions until you feel comfortable doing it

[copper roost]

i have 5 days left to redo linear algebra analysis and probability

[night raft]

how comfortable are you with it?

like

is there much to learn/revise or

[brazen breach]

I don't mean to discourage but um

[copper roost]

bro

im quite comfortable with probability and analysis

except some parts

[night raft]

that's fine

[copper roost]

like hypothesis tests, or parameter stuff

and linear algebra omg

i know like 5%

and in addition

the first partof the exam is without calculator

[night raft]

I think you can pull it off

maybe

[copper roost]

i will

[night raft]

but you'd pretty much have to neglect everything

[copper roost]

yep

idk if i should do exam questions

or just keep doing normal stuff

[night raft]

personally i'd focus exam questions

[brazen breach]

usually it helps familiarizing yourself w exam Qs especially if the prof is usually repetitive

[copper roost]

its my finals basically

id say the most difficult difficlt

like inhumane part

[night raft]

-# ok so your cooked then if it's finals

[brazen breach]

I'm ngl I'm rooting for them

[quasi rapids]

(same)

[copper roost]

is combinations in probability and hypothesis tests and parameter stuff

[quiet python]

no hes got this, 4 hours sleep every day and 20 hours study gives 100 hours of study

[brazen breach]

-# highkey impossible but like- oh no

[night raft]

lowkey don't neglect sleep

[copper roost]

nah i got this im not stupid

[brazen breach]

um

I wanna ask smth so bad

[copper roost]

yea

[brazen breach]

https://klipy.com/gifs/geometry-dash-limbo-keys--k01KQHRFYCK3MW2M3FE6SM25MX7

[night raft]

are you asking about nCr stuff

[brazen breach]

focus and lock in

[copper roost]

what does this mean

oh

[brazen breach]

I imagine it's the parenthesis thing in the distributions and some other things

[copper roost]

no like

actually yes mb

but there are like 10 formulas

[brazen breach]

that's nCr

[night raft]

it's usually for binomial distribution

[copper roost]

yeah thats easy i got that

[brazen breach]

oh wait those r the parenthesis things

[copper roost]

i had 100% on that part last exam

[untold agate]

this?

[night raft]

yeah

you can write it as nCr instead

[untold agate]

sometimes r instead of k but it doesnt matter

[brazen breach]

yeah

[copper roost]

im bad at normal distributions

and hypothesis tests

[brazen breach]

this is why I hate combinatorics and permutation we just make up notations and shi

like calculus, who even uses '

[untold agate]

i do ngl

dont gotta write d/dt n shi multiple times

[brazen breach]

how many hypothesis tests have been taught to you

[copper roost]

like 3

[brazen breach]

which ones

[copper roost]

the right left

and both

[brazen breach]

t-test?

[night raft]

You bascially want to abuse the symmetry of a normal distribution

when you are caculating probabilities

[copper roost]

yeah

nah i need atleast 45% on the exam

i got this

[untold agate]

gl bromium chip

u can close the channel if ur done btw

[copper roost]

.close

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[rotund wyvern]

hi im new to this chanel

[quiet python]

if you need help, #❓how-to-get-help

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[sterile marsh]

<@&268886789983436800>

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[unreal hamlet]

A lil confused on part c

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@unreal hamlet Has your question been resolved?

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

[unreal hamlet]

3

.close

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Please don't occupy multiple help channels.

[upper mural]

please dont open multiple channels you wont get help faster that way

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[tired osprey]

Would this how I do it pls

[loud sundial]

Sure. You could shortcut a little bit and use the Pythagorean identity directly as well instead of going to the foundations of right triangle trig.

[tired osprey]

What is that

[loud sundial]

$\sin^2 \theta+\cos^2 \theta=1$

[soft zealot]

Civil Service Pigeon

[tired osprey]

Oh

But how would I use this

[grave trout]

Sin(90-x)

[runic needle]

sin(90-x) = cos(x)

[grave trout]

And this

[tired osprey]

My teacher o my used the diagram one

[loud sundial]

I see people have been waiting to snipe

alright y'all have your fun

I'm out

[tired osprey]

[grave trout]

?

[vital crag]

don't forget to block them too

[grave trout]

I gotta go in like 5 mins

[runic needle]

?? who

[vital crag]

me

[runic needle]

u really wanna block me for saying this

[runic needle]

do you mean a right triangle?

[tired osprey]

Yeah

[runic needle]

that's valid as well

[grave trout]

I can't tell if @loud sundial wants to block me or block @runic needle

[runic needle]

wrong ping

[grave trout]

Mb

[tired osprey]

Do I end my work like that

Or is there more to be done pls

[runic needle]

what's ur final answer for sin50?

i mean, what u wrote is along the write lines

but there are inacuracies

[grave trout]

I wasn't trying to snipe. If I did, my bad

[runic needle]

how did you get x = p @tired osprey

[tired osprey]

I thought x was associated with sin

[runic needle]

he said people, not person, so i probably both of us

[tired osprey]

What is snipe

[runic needle]

dawg

do you know what right triangle we r talking about

maybe u should draw it

[tired osprey]

No

[runic needle]

sine = opp/hyp

opp = y because it's vertical

[grave trout]

Probably. I rally didn't mean to, I was just confirming that sin(90-x)=cosx in my book. Pls forgive me @loud sundial

[runic needle]

i think if you keep pinging him, it could make matters worse

[grave trout]

Well i tried

[rugged merlin]

No helper 'owns' the channel, so it's not that deep really

I think it is light etiquette

[tired osprey]

Oh

So was I finding x instead

[grave trout]

Ye but he seems pretty mad

[grave trout]

I suppose so

But I gtg now

[runic needle]

this doesn't seem mad to me

[rugged merlin]

I dont think they are mad

[runic needle]

riemann seemed more mad tbf

[serene hound]

yeah it says happy :D

[grave trout]

Ye

@loud sundial was not exactly mad but more annoyed

[rugged merlin]

You are, yes

[tired osprey]

Can I try again and show

[rugged merlin]

Ok

[runic needle]

u really testing it

[grave trout]

Ye waiting to see what he says

Should I DM him?

[runic needle]

dawg

no

just let it be

[grave trout]

Fine fine. But I rlly shldnt be texting now

[runic needle]

^

[grave trout]

Whats the worse that can happen

[runic needle]

??

[rugged merlin]

Youre just cluttering this person's help channel by this point

[grave trout]

Ur right mb

[grave trout]

I'll stop now

[wary juniper]

Now you just find x/r

[rugged merlin]

You should not be writing sin(50) = p/1

[wary juniper]

That's p/r

[rugged merlin]

The problem gave you sin(40) to work

[tired osprey]

Why

[runic needle]

what is the angle you have labelled in the triangle

40 or 50

[rugged merlin]

r = 1 so works both ways i guess

[tired osprey]

I’m confused

[runic needle]

i think decide this first then proceed

[neon cosmos]

I have a question why do you take long time at this question you can approach with identity sin^theta+cos^theta =1 then approach with formula sin(90-theta) that's simple

[wary juniper]

That's exactly what they're doing gang

[runic needle]

read the channel twin

[rugged merlin]

Simple for you. Not for her

[neon cosmos]

Ok broo chillin

[rugged merlin]

Not everybody knows this by default

[vital crag]

can you stop invading other people's channels with insulting comments. thanks

[modest sequoia]

yall let one person help out, having 3-4 people trying to say different things is overwhelming for the student

[rugged merlin]

Agreed. Ill stop

[runic needle]

i'll stop as well

[neon cosmos]

Because it's become too much messy for her

And sorry i do not insult anyone in channels

[runic needle]

we could let op decide who she wants to continue helping

[neon cosmos]

Ok

[tired osprey]

Do I change the 50 to 40 first

but how

Using 90 will change it to cos

[rugged merlin]

It's a good idea to visualise it. Can you draw a triangle real quick with edges x, y, and r and the angles 90, 40, and 50?

[tired osprey]

Ok

[rugged merlin]

Ok

[tired osprey]

Do I always draw a triangle

[rugged merlin]

So. What is sin(50) in terms of x, y, and r only?

[tired osprey]

Y over r

[rugged merlin]

Awesome

So

What about sin(40)?

Again, only in terms of x, y, and r

[rugged merlin]

Yes you should

[tired osprey]

X over r

[rugged merlin]

Yeah ok awesome

Cos(50) and cos(40)?

[tired osprey]

Y over r and x over r

But what does this do

[rugged merlin]

Alright lets notice something

[tired osprey]

I can understand doing the sin but not the cos

[rugged merlin]

Cos(40) is y over r

Sin(50) is y over r, as well

Likewise,
sin(40) = cos(50)

[rugged merlin]

Can you explain?

[tired osprey]

The questions involves sin 40 and sin 50

But idk why we’re doing this but cos

[rugged merlin]

I am just trying to explain to you the relationship between all of them with x, y, and r since you were confusing that primarily

[tired osprey]

Ohh

[rugged merlin]

Basically,
you now know that sin(40) = x/r and sin(50) = y/r

Since you are given sin(40), you should have x = p (since r = 1)

And then do the same steps you did already

[tired osprey]

I don’t get it

[rugged merlin]

Which part

sin(40) = p = x/r = x/1 = x

[tired osprey]

This

The second sentence

[vital crag]

maybe it'd help if you wrote out sin(50) and cos(50) in terms of x, y, and r. then the same for sin(40) and cos(40) to see what lex is saying

[rugged merlin]

She already did haha

[rugged merlin]

sin(40) = p, right?

Your problem says it in thr first sentence

[tired osprey]

Yes

[rugged merlin]

ok

[rugged merlin]

Here

You also said sin(40) = x/r

Agreed?

[tired osprey]

Yes

[rugged merlin]

so

sin(40) = p
sin(40) = x/r

This means p = x/r, right? @tired osprey

[tired osprey]

Yes

[rugged merlin]

r = 1, agreed?

[tired osprey]

Yeah to make it a fraction, I think

[rugged merlin]

Idk what that means but r = 1 because fhis is a unit circle

[tired osprey]

So p over 1

[rugged merlin]

A unit circle has 1 for its radius

Yeah

[tired osprey]

Oh

[rugged merlin]

So x = p/1

Or, x = p

[tired osprey]

Uhuh

[rugged merlin]

Do you still not get that second sentence now?

[tired osprey]

I get it now

[rugged merlin]

Nicee

[tired osprey]

So then I find y?

[rugged merlin]

Yuh

[tired osprey]

can I do it an show you

[rugged merlin]

You already did it!

[rugged merlin]

The work here is pretty kuch correct, its judt that your variables were assigned incorrectly

But you csn still show me

[tired osprey]

Ty

But idk how it answers the question

Bc the degrees are diff

[rugged merlin]

Degrees?

[tired osprey]

Angles

[rugged merlin]

Wdym

Reread the conversation if you must but I feel like I shared everything i could share

[tired osprey]

Ok

Ty lex

.close

[final saddle]

Channel closed

Closed by @tired osprey

Use .reopen if this was a mistake.

[rugged merlin]

Btw are you the same skittle

Or is this like a very strange coincidence lol

[final saddle]

Available help channel!

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[sterile marsh]

19

[sterile marsh]

,rccw

[sterile marsh]

!status

[final saddle]

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

[sterile marsh]

1

.close

[final saddle]

Channel closed

Closed by @sterile marsh

Use .reopen if this was a mistake.

[worldly spruce]

Interesting

Asked the question, rotated it, stated the status, and solved it themselves

[lofty sinew]

They just needed a new perspective

[worldly nymph]

ayo man

[worldly spruce]

I know, it was a joke

[final saddle]

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
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• Type the command .close to free the channel when you're done.
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Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[raw patrol]

Hiii

[raw patrol]

Im confused with proof by induction

I’ll show my question and working

[vital crag]

use k! = k * (k-1)! and this

you're supposed to start with induciton hypothesis, not the conclusion from induction

[rocky tusk]

it might help to take the inequality you want to arrive at then reduce it to something you know is true

[raw patrol]

How do I do that

In the bottom left corner I wrote where I’m trying to get to

Oh wait

[rocky tusk]

well you want to get to
(n + 1)! > (n + 1)^2 + (n + 1) yes?

[raw patrol]

It should be +2

[raw patrol]

Yeah

[rocky tusk]

wdym

oh i see in the bottom left

yes

[raw patrol]

In my working bottom left corner I forgot +1 th

Yh

[rocky tusk]

yea you forgot to make the last part k + 1

[raw patrol]

Yh

But regardless how do I comapre

[rocky tusk]

was this middle bit you trying to get the inequality?

[raw patrol]

Yeah

[rocky tusk]

it looks like you multiplied by k + 1

let me latex this

$(k + 1)! > k^2(k + 1) + k(k + 1)$

[raw patrol]

I did so k! Goes to (k+1)!

[soft zealot]

knief

[rocky tusk]

mhm

[raw patrol]

Yh

[rocky tusk]

now you want the right side to be (k + 1)^2 + (k + 1). we can factor to get k(k + 1)^2 like you did

we wanted to show it’s >= (k + 1)(k + 2) right

or

(k + 1)^2 + (k + 1)

[raw patrol]

Yesss

[rocky tusk]

so you can simply divide both sides by k + 1 to get k(k + 1) >= k + 2

k^2 - 2 >= 0

which holds since k >= 4

[raw patrol]

What

[rocky tusk]

ok i’ll show you in latex

we have $(k + 1)! > k(k + 1)^2$. we want to then show that [k(k + 1)^2 \geq (k + 1)^2 + (k + 1) = (k + 1)((k + 1) + 1) = (k + 1)(k + 2)] since $k \geq 4$ we can simply divide both sides by $k + 1$ to get [k(k + 1)^2 \geq (k + 1)(k + 2) \iff k(k + 1) \geq k + 2 \iff k^2 + k \geq k + 2]

[soft zealot]

knief

[rocky tusk]

then subtract k from both sides to get k^2 >= 2

which is obviously true since k >= 4

if you aren’t comfortable with why we can divide by k + 1 then lmk

[raw patrol]

Oh yeah I understand what you did

I just wouldn’t think of that

[rocky tusk]

takes practice

if i see something that i can factor, i factor it

making it (k + 1)(k + 2) helped

[raw patrol]

Yep thx

[rocky tusk]

no worries

like your original statement is really just (n - 1)! > n + 1

for n >= 4

divide by n

[raw patrol]

Yeah true

Hm

I have another kinda similar question too

Again like idk how to prove that 2k^2>(k+1)^2

[rocky tusk]

i think this is supposed to n^2

[raw patrol]

Oh

Yeah

[rocky tusk]

can you show your work

oh you just

[raw patrol]

I skipped the base case but yh

[rocky tusk]

multiplied by 2

[raw patrol]

Yep

[rocky tusk]

well try expanding this

collect like terms

[raw patrol]

Err

Am I doing this right?

[rocky tusk]

nice

so you’re trying to prove that for k >= 5,
k^2 >= 2k + 1

[raw patrol]

Ohh

[rocky tusk]

there are a few ways you can do this

[raw patrol]

Why do you include the =

[rocky tusk]

because for transitivity we don’t need a strict >

if a > b and b >= c then a > c

sometimes it matters

[raw patrol]

I see

[rocky tusk]

in other problems you might not get b > c

it might be b >= c

so i just make it a habit to use >=

even if equality doesn’t hold

anyways

do you have any ideas?

you can do this with algebra

or induction if you really want

[raw patrol]

Bring it over to one side and complete the square ?

[rocky tusk]

nice

[raw patrol]

Is this ok

[rocky tusk]

yea

i don’t know how strict your graders are

maybe they want you to show clearly why (k - 1)^2 >= 2 for k >= 5

but i’d accept that

[raw patrol]

How would I do that

[rocky tusk]

just so you know,

yea

ok

k >= 5 implies that k - 1 >= 4 so
(k - 1)^2 >= 16 and so (k - 1)^2 - 2 >= 14 >= 0

easy enough right

these sort of statements that involve just simple expressions in terms of k can often just be manipulated/derived from the starting condition on k (here it’s k >= 5)

[raw patrol]

Cool this has been rlly useful

[rocky tusk]

always keep in mind what we’re assuming is true

[rocky tusk]

k >= 5 here is sort of our "base"

it’s the statement we take to be true then discover other true statements that follow from it

this is what math is about

given a certain set of true statements, what else is true if those are?

[rocky tusk]

like in a sense, these are the same statements just in a different form

maybe not "same"

since they don’t both imply each other

but k >= 5 implies the other one

[final saddle]

@raw patrol Has your question been resolved?

[final saddle]

Channel closed

Closed due to timeout

Use .reopen if this was a mistake.

[final saddle]

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• Do not immediately ping people or roles. After 15 minutes, feel free to ping <@&286206848099549185> once.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[rocky tusk]

what do you think

[gentle zephyr]

i am not sure

[rocky tusk]

have you at least tried some examples

as in, have you tried some values of a and b

and found the gcd and lcm

[gentle zephyr]

i want to know why

[rocky tusk]

i will explain

but it doesn't hurt to have a good idea of what the answer is first

[gentle zephyr]

d = gcd(a,b) => d | a , d | b => a = d.k , b = d.q

[rocky tusk]

have you seen any other definitions?

perhaps involving the prime factorization

[gentle zephyr]

l = lcm(a,b) => ?

[gentle zephyr]

like which one

[rocky tusk]

like gcd takes the minimum exponent from each prime power in the prime factorizations of a and b

https://www.math.cmu.edu/~cnewstea/teaching/old/teaching/21-127-S14/handouts/gcd-with-fta.pdf

for lcm we take delta_i = max(...)

[gentle zephyr]

isnt it complicated

[rocky tusk]

no this is really quite simple actually

this is how you'd probably go about finding the gcd in your head in elementary school

have you not seen this before?

the proof i have in mind involves these definitions

[rocky tusk]

yes this is usually how they first define it

[rocky tusk]

usually you come across this definition shortly after

which imo is a better definition

[gentle zephyr]

i mean intuitively i understand this def

thats what u do in practice

[rocky tusk]

it literally turns into proving min(n,m) + max(n, m) = n + m

[rocky tusk]

mhm

[gentle zephyr]

?

[rocky tusk]

to prove the statement we want to end up proving we just need to prove this

do you know which answer choice is correct btw

or at least have any guesses

[gentle zephyr]

D of course

[rocky tusk]

so if we write [a = p_1^{\alpha_1} p_2^{\alpha_2} \cdots p_n^{\alpha_n}, \quad b = p_1^{\beta_1} p_2^{\beta_2} \cdots p_n^{\beta_n}] then what does $ab$ equal?

[soft zealot]

knief

[rocky tusk]

in terms of the prime factorization

[gentle zephyr]

alfa + b

[rocky tusk]

mhm

so

$ab = p_1^{\alpha_1 + \beta_1} p_2^{\alpha_2 + \beta_2} \cdots p_n^{\alpha_n + \beta_n}$

[soft zealot]

knief

[gentle zephyr]

y

[rocky tusk]

now

[rocky tusk]

are you saying and or why

[gentle zephyr]

ye

[rocky tusk]

ok i'll assume you mean and

[gentle zephyr]

i Meb yes

[rocky tusk]

now we need to find the prime factorization of gcd(a,b) lcm(a, b)

[gentle zephyr]

mhm

[rocky tusk]

i'll use this notation ok

also let's say uhh varphi is max

like $\varphi_i = \max(\alpha_i, \beta_i)$ where $1 \leq i \leq n$

[soft zealot]

knief

[rocky tusk]

can you tell me what gcd(a, b) lcm(a, b) is

like write the prime factorization

you don't have to write it entirely i guess

just in words

[gentle zephyr]

wdym

[rocky tusk]

i wrote the prime factorization for ab

[gentle zephyr]

gcd = p1^phi

[rocky tusk]

can you do that for gcd(a, b) lcm(a, b)

[gentle zephyr]

gcd = p1^min(alf1, veta1)

@rocky tusk

[rocky tusk]

yea

and lcm is max

so what about the product

[gentle zephyr]

?

[rocky tusk]

if we multiplied them what would we get

[gentle zephyr]

?

[rocky tusk]

no we used delta for min

[gentle zephyr]

how do u know

@rocky tusk

[rocky tusk]

wdym how do i know

i thought we agreed on this 15 minutes ago

[gentle zephyr]

no

[gentle zephyr]

this definition is for gcd

[rocky tusk]

you're ok with this though?

[gentle zephyr]

what about lcm

[rocky tusk]

replace min(alpha, beta) with max(alpha, beta)

and you get lcm

we take the highest power of each prime

because we need it to be a multiple

for example the lcm of 2 * 3^3 * 5^2 * 7 and 5^3 * 7^2 * 11 is 2 * 3^3 * 5^3 * 7^2 * 11

we need it to be a multiple of each so it has to have at least as many copies of each prime from both

and since we don't add anymore copies of the primes we get the least common multiple

does this feel intuitive

for the gcd we needed it to divide both so we couldn't have more copies of any particular prime than one of the two numbers had individually, so we would take the smallest prime power for each prime

[gentle zephyr]

what is a multiple

[rocky tusk]

a is a multiple of b is a = bk for some integer k

12 is a multiple of 4 since it's 3 times 4

[gentle zephyr]

a | b <=> b = a.k

[rocky tusk]

yes notice the difference

a being a divisor of b means b is a multiple of a

[gentle zephyr]

a being a multiple of b means a is a divisor of b

[rocky tusk]

on the one hand this says that a divides b, but on the other hand it says that b is a multiple of a

[rocky tusk]

other way around. it means b is a divisor of a

[gentle zephyr]

how?

[rocky tusk]

4 divides 12

12 is a multiple of 4

have you never seen the term "multiple"

[gentle zephyr]

i have heard

[rocky tusk]

so you agree with this?

[rocky tusk]

and this

what do you normally say in spanish?

back when you were a kid learning about this stuff

[gentle zephyr]

multiplo

minimo comun multiplo

lcm

[rocky tusk]

sounds like the same thing to me

😭

i could've guessed that

so what's confusing

[rocky tusk]

you agree with this?

[gentle zephyr]

yeah um just bad at mwth

[rocky tusk]

the variables make it confusing

i hope we can agree that 12 is a multiple of 4 and 4 is a divisor of 12

[gentle zephyr]

is just max×min

[rocky tusk]

wdym

[gentle zephyr]

surely

[rocky tusk]

so this makes sense

[gentle zephyr]

ab = p^(max x min

[rocky tusk]

well we said ab = p^(alpha + beta)

we are trying to figure out what gcd times lcm is

we had a detour

now do you agree that lcm(a, b) = p_1^(max(alpha, beta)) ....

[rocky tusk]

like did this make sense

[gentle zephyr]

is p^min x p^max = p^min+max

[rocky tusk]

yes

so we get p^(min + max)

and we are trying to show that it's equal to p^(alpha + beta)

so we simply show that min(alpha, beta) + max(alpha, beta) = alpha + beta

try this

lmk how it goes

[gentle zephyr]

how

[rocky tusk]

don't overcomplicate it

[gentle zephyr]

what if min = max

[rocky tusk]

you tell me

[gentle zephyr]

ah ten a+b=2a=2b

[rocky tusk]

yep

[gentle zephyr]

ye

[gentle zephyr]

three cases a < b , b > a and a = a

[rocky tusk]

yea or you can just say WLOG

and do one case

[gentle zephyr]

min = a or min = b or min = a = b

[gentle zephyr]

care to elaborate

[rocky tusk]

well the cases alpha <= beta and beta <= alpha are identical

it's just a difference in labeling

we can just say without loss of generality assume alpha <= beta

[gentle zephyr]

how

[rocky tusk]

we are trying to show min(alpha, beta) + max(alpha, beta) = alpha + beta for arbitrary alpha and beta

alpha and beta are just any nonnegative integers in this context

they are just placeholders

like ok if you prove it for a < b

can't we just relabel a and b

a --> b and b --> a

the labels themselves are meaningless

integer one and integer two

[gentle zephyr]

wlog the proof is that min + max = a + b

[rocky tusk]

yes

but wlog assume a <= b

maybe it would be instructive for you to do the extra cases so you can see what i mean

[gentle zephyr]

a < b
max + min = b + a = a + b the field Z is commutative under addition

a > b
mac + min = a + b ✅

a = b
a + b = 2b

✅

[rocky tusk]

easy enough

[gentle zephyr]

so p^min + max = p^ a + b

[rocky tusk]

yep

[gentle zephyr]

so ab = gcd x ñcm

[rocky tusk]

enya

[gentle zephyr]

ñ = L

[rocky tusk]

does it

[gentle zephyr]

≡

i appreciate the heñp

[rocky tusk]

lol

you're welcome

[gentle zephyr]

.soñved

.cñose

.close

[final saddle]

Channel closed

Closed by @gentle zephyr

Use .reopen if this was a mistake.

[gentle zephyr]

my keyboard, dayum

[rocky tusk]

i have to hold the n

for the accent

[gentle zephyr]

[rocky tusk]

n

[final saddle]

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• Do not immediately ping people or roles. After 15 minutes, feel free to ping <@&286206848099549185> once.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[raw patrol]

Hi I’m a bit confused on the second / third part

[raw patrol]

I can show my working so far

[runic needle]

what did you get for part i

peace out

[gritty chasm]

I may drop a hint for the first one(you've already solved it, this is for future problems): if $z$ is a root of a polynomial with real coefficients, then $\bar{z}$ is also a root.

[soft zealot]

Annie Maqionde

[gritty chasm]

For the second part well you do know that:\

1. sum of roots = $-a$\
2. product of roots = $b$, right?

[soft zealot]

Annie Maqionde

[gritty chasm]

For the third one you may reduce it to a quadratic by an appropriate subsitution, and then go forth and square root the roots of the resulting quadratic.

[final saddle]

@raw patrol Has your question been resolved?

[raw patrol]

Ohh

I overcomplicated

It’s just 16+30i

[raw patrol]

Yeah my teacher also told us x^2-2Re+|z^2|

[raw patrol]

Hm

I still don’t fully get iii

Is it this

[late gazelle]

is this for part iii?

[raw patrol]

Yh

[late gazelle]

so $y^4 + ay^2 + b = 0$

[soft zealot]

KB

[late gazelle]

we dont know how to solve a general degree 4 polynomial (without massive formula) so instead lets make a substitution

let V = y^2

what do you notice about the new equation?

[raw patrol]

It’s a quadratic that we know the roots to

[late gazelle]

mhm

[raw patrol]

V^2-32V+1156=0

V1 = 16-30i and V2=16+30i

[late gazelle]

mhm, now V = y^2 right

so lets sub y^2 in for V and we get our answers

[late gazelle]

so your process is correct

so what in particular doesnt make sense

[raw patrol]

It’s 7 marks

I feel like I’m missing something

[late gazelle]

its just asking for the 4 roots to the equation no?

for my complex numbers test the other year a similar question was worth 8 marks i believe

[raw patrol]

Also in the markscheme they got + or - (5 + or - 3i)

[late gazelle]

thats just simplifying the square root

[raw patrol]

When I simplified it I got a different answer

[late gazelle]

let me quickly go over the question

[raw patrol]

Ok thx

[late gazelle]

what were the values you got for a and b

[raw patrol]

-32 and 1156

[late gazelle]

hmm

i got the same answers you got

are you sure a and b are correct?

[raw patrol]

Yeah

[late gazelle]

ok nvm

what you have is correct, its just not fully simplified

[raw patrol]

The main reason I’m confused rn is bc they got 5+3i whilst I got 5-3i for my root

[late gazelle]

this is what I got: $y = \pm \sqrt{-\frac{a}{2} \pm \sqrt{\frac{a^2}{4}-b}}$

[soft zealot]

KB

[late gazelle]

although, it may be that its pm, and then mp