As FreddieMercuryFan mentioned, instead of base-10, this is a base-3 system.
#help-39
minor eagle
yes
humble root
So, the first thing you should do is tell me what the powers of 3 are :) (The first four powers will do)
minor eagle
something like,
3 0 = 1
3 1 = 3
3 2 = 9
3 3 = 27
?
blazing garden
something like,
3 0 = 1
3 1 = 3
3 2 = 9
3 3 = 27
?
Yessir!!!
And how many of each do you have?
How many:
3^3
3^2
3^1
3^0?
minor eagle
uhh
humble root
(^ I.e, how many blocks of 27 can you see?)
blazing garden
Right idea, wrong numbers
minor eagle
i figured
blazing garden
There is 1 block of 27
And 2 squares of 9
Do you see where I’m going with this?
minor eagle
i believe so
blazing garden
i believe so
Ok, how many 3^1 groups are there?
minor eagle
0?
blazing garden
What is 3^1?
blazing garden
Ok, so… how many blocks/lines of 3 are there?
minor eagle
is it not 0
blazing garden
is it not 0
There are 3
minor eagle
sorry its just very jumbling for me
blazing garden
sorry its just very jumbling for me
Listen man, no need to apologize
I’m gonna walk you through every step. No judgment
You’re all good broski 😎
minor eagle
Ok, so… how many blocks/lines of 3 are there?
ok so 3
minor eagle
5?
minor eagle
well isnt 3 0 just a singular cube
blazing garden
well isnt 3 0 just a singular cube
Yep. And I think there’s a bit more than 5 of them
minor eagle
basically what i did was
1 block of 27
Contains 27 units
2 blocks of 9
Each contains 9 units
So together:
9+9=18
3 blocks of 3
Each contains 3 units
So together:
3+3+3=9
Loose single cubes
There are 6 of them
27+18+9+6=60 units (60 blocks of 3^0)
blazing garden
1 block of 27
Contains 27 units
2 blocks of 9
Each contains 9 units
So togeth...
And that is correct. But that’s different than what you’ve told me
3³ : 1
3² : 2
3¹ : 3
3⁰ : 6
minor eagle
i have to tweek a bit
blazing garden
i have to tweek a bit
It’s all good bro.
But 60 is correct
And now, we have to convert 60 into base 3
Do you need help with that?
minor eagle
i got 2020^3
blazing garden
to recap 2020₃ = 60₁₀ is the final answer
YESSIR!!!
pearl pondBOT
Available help channel!
royal galleon
can someone explain why 2n+1? say we had n = 5. so we would have 1-x^2+x^4-x^6+x^8-x^10 + (-x^2)^6/1+x^2 = 1-x^2+x^4-x^6+x^8-x^10+x^12/1+x^2. But that is 2n+2
worldly jewel
typo probably
pearl pondBOT
pearl pondBOT
Available help channel!
acoustic tangle
pearl pondBOT
pearl pondBOT
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
@acoustic tangle Has your question been resolved?
pearl pondBOT
acoustic tangle
.reopen
pearl pondBOT
✅ Original question: #help-39 message
acoustic tangle
!status
I got an answer and it’s wrong
I thought it was the one of t he very right bc it’s median is low as the graph is negatively skewed but it’s not
paper terrace
I got an answer and it’s wrong
Could you tell me what the value of the median is?
pearl pondBOT
@acoustic tangle Has your question been resolved?
pearl pondBOT
acoustic tangle
.reopen
pearl pondBOT
✅ Original question: #help-39 message
sage mortar
yo
acoustic tangle
Could you tell me what the value of the median is?
Im not sure bc there are no numbers
paper terrace
hi
you have 8 integer values right?
what would you say is the median of that 8 values
pearl pondBOT
@acoustic tangle Has your question been resolved?
fast hill
hi
acoustic tangle
what would you say is the median of that 8 values
4.5
acoustic tangle
In the middle
acoustic tangle
Like the average value
paper terrace
Okay, so, when will the average value be higher?
paper terrace
okay, so, when the rectangles are mostly on the right, the mean is higher right?
acoustic tangle
Yep
paper terrace
looking at the graphs, when are the rectangles on the right bigger the median rectangle?
acoustic tangle
Not sure
paper terrace
Not sure
sorry for the delay
Why are you not sure
Imagine the median rectangle on every graph
and look at the right of it
make this median rectangle with the average size between the rectangle on the left and the one on the right
acoustic tangle
Hm
acoustic tangle
looking at the graphs, when are the rectangles on the right bigger the median re...
I don’t get ur question tbh
paper terrace
in those graphs
in which graph are the right rectangles bigger than the median rectangle
pearl pondBOT
@acoustic tangle Has your question been resolved?
pearl pondBOT
pearl pondBOT
Available help channel!
torpid wolf
this really shouldnt have been hard but
pearl pondBOT
this really shouldnt have been hard but
torpid wolf
first principles is lim(h->0) of (f(x+h)-f(x))/h
but the issue is in this case it's
(root(x+h) - root(x))/h
lim(h->0) of root(x+h) is root(x)
so i get 0?
which doesnt make sense
prisma kernel
Have you tried multiplying with its conjugate
torpid wolf
how on earth do i get the (1/2root(x))
torpid wolf
Have you tried multiplying with its conjugate
wdym
prisma kernel
Do you know what a conjugate is?
torpid wolf
Do you know what a conjugate is?
in complex numbers yeah
but i havent heard it much with real numbers
but i see what you mean
torpid wolf
so you mean completing the square
no not that sorry
prisma kernel
Basically for something like (A + B), it's conjugate is (A - B)
torpid wolf
difference of two squares kind of
torpid wolf
Basically for something like (A + B), it's conjugate is (A - B)
yeah thats what i meant kinda
so you're saying i multiply it so i get
prisma kernel
You multiply and divide with a conjugate, it's a fairly known strategy for limits
torpid wolf
No no
on the numerator it's fine it'll cancel out
denominator i will get h(sqrt(x-h))
prisma kernel
You're doing this wrong mate
prisma kernel
You had (sqrt(x + h) - sqrt(x))/h
You had this
torpid wolf
You had this
yes
and you said to multiply both numerator and denominator by sqrt(x-h)?
prisma kernel
You multiply and divide with (sqrt(x + h) - sqrt(x))
prisma kernel
Basically for something like (A + B), it's conjugate is (A - B)
Oh no, use this for reference
torpid wolf
oh
prisma kernel
Basically for something like (A + B), it's conjugate is (A - B)
Here you can replace A with sqrt(x + h) and B with sqrt(x) to get our original function
torpid wolf
i realised why
you meant A as sqrt(x+h) and B as sqrt(x)
when you said A+B vs A-B as conjugates
i thought for some reason that you wanted me to say A was x and B was h
and then do the sqrt's that way
my bad
prisma kernel
i thought for some reason that you wanted me to say A was x and B was h
No no, well It's fine that you get it now
Let me know what you get
torpid wolf
Let me know what you get
ok
should be pretty okay now im surprised i didn't know this technique with limits
prisma kernel
should be pretty okay now im surprised i didn't know this technique with limits
Well since you've just entered derivatives with first principle it's fine
There are a lot more
prisma kernel
Alright then🫡
torpid wolf
Well since you've just entered derivatives with first principle it's fine
i learned first principles like 2 years ago
lmao
but tbf i dont use it much
prisma kernel
Oh damn, yeah nobody does TBF
torpid wolf
i did still do a lot of limits though with other things
timber cape
-# foreshadowing
torpid wolf
so idk
prisma kernel
Well all the best for your future questions man
pearl pondBOT
pearl pondBOT
Available help channel!
long nova
how does he get the answers
pearl pondBOT
how does he get the answers
broken ivy
<@&268886789983436800> get him
paper terrace
<@&268886789983436800> get him
hahahahahah
long nova
<@&286206848099549185>
thorny oasis
it can't be the real deal
left steppe
!15m
pearl pondBOT
Please only use the <@&286206848099549185> ping once if your question has not been answered for 15 minutes. Please do not ping or DM individual users about your question.
paper terrace
it can't be the real deal
double channel + pinging helpers + impersonating isaac newton
thorny oasis
double channel + pinging helpers + impersonating isaac newton
triple ban def
pastel umbra
how does he get the answers
I mean, it just falls out of trig
thorny oasis
how does he get the answers
its at the top half of your screen
thorny oasis
I mean, it just falls out of trig
(he's alr timed out I believe)
pastel umbra
(he's alr timed out I believe)
? No, I think the person who was banned was the impersonator of the user who bears the name Sir Isaac Newton
Not of the English mathematician
oh wait nvm
...No the ban itself was something else; there's been an influx of cryptoscam messages on Discord lately
thorny oasis
oh, i see...
long nova
I mean, it just falls out of trig
bro why is unit vector not 1/7(2 3 -6)
i thought unit vector was OP vector / magnitude of OP
pearl pondBOT
broken ivy
hahahahahah
idk why you having a crashout in a help channel but I called mods because there was a scammer here, not because the guy is called isaac newton
paper terrace
idk why you having a crashout in a help channel but I called mods because there ...
It was because he opened two channels, not a big of a deal
pearl pondBOT
Available help channel!
maiden crest
<@&268886789983436800>
alpine cipher
pearl pondBOT
alpine cipher
hello
can someone pls help with part a and c
ping me if someone comes
for part a, i dont get why ln is negative
summer imp
for part a, i dont get why ln is negative
The limit of the fraction inside the logarithm is $\frac{8}{10}$, so $$\lim_{n\to\infty} \ln\left(\frac{8n^7 + 10}{10n^7 +8}\right) = \ln\left(\lim_{n\to\infty}\frac{8n^7 + 10}{10n^7 +8}\right) = \ln\left(\frac{8}{10}\right).$$
jolly parrotBOT
Azyrashacorki
summer imp
And -ln(10/8) = ln(8/10)
alpine cipher
oh
brave sluice
yes that's the rule about powers inside limits
alpine cipher
okay cuz i kept getting ln8/10 but the answer was -ln10/8
so i got confused 😭
i forgot they were the same
summer imp
It's unfortunate that they didn't accept it, but they are the same yes
summer imp
Hum while the limit of what's inside arctan goes to infinity, arctan has a limit as its argument goes to infinity
alpine cipher
which i divided by n
alpine cipher
Hum while the limit of what's inside arctan goes to infinity, arctan has a limit...
oh
so pi/2?
summer imp
Yep
pearl pondBOT
pearl pondBOT
Available help channel!
sharp smelt
pearl pondBOT
timber cape
oh damn congrats on va
timber cape
uncongrats on va
sharp smelt
So to start, $p= \frac{\abs{\bar{x}-\mu}}{\sigma_{0}/\sqrt{n}}$
jolly parrotBOT
Wai
sharp smelt
\bar{x}= 4.88 so p=0.07589466384
so we have sufficient evidence to believe H_0 is true
hmm, a 0.95 confidence interval for $\mu$ now
jolly parrotBOT
Wai
pearl pondBOT
pearl pondBOT
Available help channel!
ebon burrow
pearl pondBOT
hard relic
What have you tried?
ebon burrow
I wrote down the first 20 terms of the sequence
And after that I couldn't do anything
hard relic
I wrote down the first 20 terms of the sequence
That's not going to help you prove anything lol
ebon burrow
Ikr
wooden flare
do you know what form you should end up with ?
ebon burrow
But did it for some intuition
ebon burrow
do you know what form you should end up with ?
Nope
I thought of trying to solve the recurrence
But then it's prolly unsolvable
ebon burrow
Try induction
Uhm but how exactly? Do I show that if 2^k+k-2 is a square then 2^(k+1)+k-1 is a square as well?
hard relic
Uhm but how exactly? Do I show that if 2^k+k-2 is a square then 2^(k+1)+k-1 is a...
No. That has nothing to do with what has been asked
ebon burrow
No. That has nothing to do with what has been asked
Then on what do I induct?
hard relic
n is your index. The perfect quare is the element of the sequence for said form of index
ebon burrow
n is your index. The perfect quare is the element of the sequence for said form ...
Soo I induct on n?
But how exactly
hard relic
Soo I induct on n?
Hint: a_n is the k^th perfect square in the sequence.
ebon burrow
Hint: a_n is the k^th perfect square in the sequence.
so i show that my k+1th perfect square has an index of the same form right>
?
wooden flare
$a_n = a_k - \floor{\sqrt{a_n}}$ where $k \in Z+$ and $k=n+1$, we know $a_n$ should end up being a perfect square thus, we prove that $a_k - \floor{\sqrt{a_n}}$ can be written as J^2 where $J \in N$ or at least that should be something
jolly parrotBOT
KB
Compile Error! Click the 
reaction for more information.
(You may edit your message to recompile.)
ebon burrow
**KB**
Compile Error! Click the <:errors:504266556054962196> reaction for more i...
Ok
wooden flare
take mine with a grain of salt btw
timber cape
what
hard relic
Ok lemme try and report back
It would help to remove the floor function from the recurrence first.
ebon burrow
take mine with a grain of salt btw
Ok
ebon burrow
It would help to remove the floor function from the recurrence first.
How would one do that
Like could you explain pls
I always wonder can we solve such recurrences even?
hard relic
How would one do that
Think about what happens to the floor function for a perfect square's root.
ebon burrow
Think about what happens to the floor function for a perfect square's root.
It simply equals √x_n
Ok I am trying the problem please don't mind if I don't respond for a bit 😭
hard relic
Ok I am trying the problem please don't mind if I don't respond for a bit 😭
It's alright. Try it with what you know for now. We'll see how it goes.
pearl pondBOT
pearl pondBOT
Available help channel!
pearl pondBOT
Available help channel!
minor cloud
<@&268886789983436800>
pearl pondBOT
pearl pondBOT
Available help channel!
lusty pier
hey can you help me understand what a cartesian is
pearl pondBOT
hey can you help me understand what a cartesian is
hoary relic
hey can you help me understand what a cartesian is
Cartesian what?
lusty pier
smoky musk
Maybe it's a vector space represented by $\mathbb{R}^n$?
jolly parrotBOT
Xwtek
lusty pier
hm maybe
i just kind of forgot what cartesian were
like i assume cartesian coordiantes
isnt it x,y to x,y,z
smth liek that
primal stump
isnt it x,y to x,y,z
you can interpret it as an equation for different figures in space like a straight line a field and a sphere
for example you got the cartesian equation for a sphere is : (x-a)² + (y-b)² + (z-c)² = R²
i dont know if this is the right answer for you question
hope this helps
ashen ivy
i don't want to give too much away, but Cartesian space has co-ordinates x,y,z (corresponding to length/width/height of an object), basically it measures how much space the object takes up. k-space has different co-ordinates, k_x, k_y, k_z, which represent a kind of frequency of the object
so what you're looking for is some kind of transformation that goes from a spatial domain to a frequency domain
pearl pondBOT
@lusty pier Has your question been resolved?
lusty pier
ig its kind of confusing on what you meant by "how much space the object takes up" bc is each pixel has some sort of frequency
pearl pondBOT
pearl pondBOT
Available help channel!
winged pulsar
Did i write it correctly?
pearl pondBOT
Did i write it correctly?
main oriole
tg(x) is portuguese tangent in this case correct?
winged pulsar
tg(x) is portuguese tangent in this case correct?
Yes but it’s polish tangens
main oriole
yea those are similar
winged pulsar
Ok is it good write?
modern talon
whats portugeuse tangent?
the portuguese denote tangent as tg(x) rather than other languages that denote it as tan(x)
modern talon
Did i write it correctly?
anyways, does thatsay tan(v + h pi) = tan(v)?
haughty lynx
the portuguese denote tangent as tg(x) rather than other languages that denote i...
oh i thought it meant something else altogether haha
modern talon
or are you just asking if tan(60) is root 3 lol
haughty lynx
anyways, does thatsay tan(v + h pi) = tan(v)?
if h is an integer then thats correct innit
winged pulsar
anyways, does thatsay tan(v + h pi) = tan(v)?
Yes
haughty lynx
what exactly is your doubt then
if youre asking about your handwriting its terrible yes
otherwise whatever youve written is correct
dusty jungle
if youre asking about your handwriting its terrible yes
its lowkey good though
modern talon
Is this correct?
looks good
modern talon
its lowkey good though
pause
alpine cipher
pearl pondBOT
cursive wraith
So you managed to find the formula for $\sum_{n=2}^{\infty}(n^2-n)x^n$
jolly parrotBOT
Rafilouyear2026
alpine cipher
yeah
alpine cipher
mhm
cursive wraith
= ?
tulip ore
given this is part (f), were you required to solve $\sum_{n=1}^\infty\frac n{3^n}$ before?
jolly parrotBOT
mtt
alpine cipher
yeah i did
tulip ore
yea
alpine cipher
thx guys 😭 🙏
tulip ore
np
pearl pondBOT
jolly obsidian
.reopen
pearl pondBOT
Available help channel!
pearl pondBOT
Available help channel!
warm elbow
Prove that if $C$ is a proper class of sets, then there cannot be an ordinal $\gamma$ and a bijection $f: \gamma \to C$. (That is, $C$ does not have a cardinality).
jolly parrotBOT
toast
warm elbow
What is the general approach to solve this problem?
Contradiction?
maybe assume C is a proper class of sets, and there is an ordinal and bijection
then show C is not a proper class?
summer imp
Visions of ||replacement||?
warm elbow
hmm
pearl pondBOT
@warm elbow Has your question been resolved?
warm elbow
Visions of ||replacement||?
is the idea that
we apply replacement on gamma
such that every element of gamma maps to an element of C
but bc is a ijection
f[gamma] = C
hence a contradictino since C is not a set?
\textit{Proof: } Assume by way of contradiction that $C$ is a proper class of sets, and there is an ordinal $\gamma$ and a bijection $f: \gamma \to C$. We will apply replacement to $\gamma$, such that we generate $A = { f(\beta) : \beta \in \gamma}$. Observe that $A$ is a set by this axiom, and because $f$ is a bijection, we get $A = C$. But this contradicts $C$ being a proper class of sets, hence such an ordinal and bijection cannot exist. $\blacksquare$
jolly parrotBOT
toast
warm elbow
<@&286206848099549185> can someone verify my logic
pearl pondBOT
@warm elbow Has your question been resolved?
pearl pondBOT
pearl pondBOT
Available help channel!
full lark
pearl pondBOT
light helm
!nopdf
pearl pondBOT
Please post images (such as PNGs or JPGs) of the question rather than other filetypes such as PDFs which have to be downloaded. Non-image downloads can potentially contain viruses or other security risks.
full lark
pearl pondBOT
@full lark Has your question been resolved?
full lark
Any novel constants or functions introduced must be definable independently of the logistic map and admit a finite axiomatic characterization.
full lark
pearl pondBOT
@full lark Has your question been resolved?
pearl pondBOT
full lark
.reopen
pearl pondBOT
✅ Original question: #help-39 message
pearl pondBOT
@full lark Has your question been resolved?
full lark
한국인이라서 한국어도 가능합니다
pearl pondBOT
@full lark Has your question been resolved?
crystal hollow
😭
crystal hollow
한국인이라서 한국어도 가능합니다
what kind of math r u studying?
full lark
I’m more of a math enthusiast than a pro
(im not good at math...),
but I’ve been really into the logistic map and chaos theory lately !
full lark
what kind of math r u studying?
.
pearl pondBOT
@full lark Has your question been resolved?
crystal hollow
I’m more of a math enthusiast than a pro
(im not good at math...),
but I’ve bee...
I see. I don't really know about this, but good luck with ur study!
full lark
There has been no proof that a generalized closed-form solution for the logistic map does not exist. (including special function)
full lark
Do you have a question
.
A generalized closed-form solution for the logistic map exists, but requires a special function not expressible in terms of known analytical functions.
I aim to characterize this special function, specifically its relationship with elementary functions and whether it can be represented in a closed form using specific constants.
pearl pondBOT
@full lark Has your question been resolved?
pearl pondBOT
pearl pondBOT
Available help channel!
outer crystal
Can someone explain this
pearl pondBOT
Can someone explain this
rough stream
fx(1,2) is the slope of the function at (1,2) when traveling in the x-direction only
So imagine starting at (1,2) and walking in the same direction as that x-axis arrow. Are you climbing up or down?
pearl pondBOT
@outer crystal Has your question been resolved?
outer crystal
oh got it
pearl pondBOT
pearl pondBOT
Available help channel!
woven isle
How do I find out what 2 numbers is sin(ax) between on any given interval, for example what 2 numbers is sin(pix) between in the interval [-1,1]
woven isle
Without desmos or any tech just paper and a brain
Ik that sin(ax) is always between 1 and -1 on R but how do I use that to my advatage
feral olive
It should be in radians given the pi but yes
woven isle
Pi/2. 3pi/2
feral olive
+2kpi though because it can cycle
Are you finding the values given in a certain range
woven isle
+2kpi though because it can cycle
Lets stay in 2pi to 0 for now
feral olive
You should still think about the cycles though because it goes from -1 to 1
feral olive
Oh its an integral
woven isle
If i find out what sin(pix) is between on -1 and 1 thats all I need
feral olive
First you integrate sinpix + 2x
woven isle
I have the answer and solution but I want to know how to get the value of sin(ax) for anything
feral olive
Then plug in 1 and -1
feral olive
But its an integral not an inequality
woven isle
They want it as one
Idk man i just want my A+
Is it readable or should I redo it
.close
pearl pondBOT
pearl pondBOT
Available help channel!
hazy quail
A round balloon of radius r subtends an angle A at the eye of the observer while the angle of elevation of its centre is B. Prove the height of the centre of the balloon is rsinBcosecA/2
honest oyster
What have you tried
hazy quail
What have you tried
i cant draw diagram and get correct answer apparently
i saw the answer and diagram, and i cant seem to understand it
honest oyster
Can you show it to us?
hazy quail
Can you show it to us?
i cant show it as of now
Due to some issue
i can show it later
oh nvm i just understood how it works
.close
pearl pondBOT
warped violet
pearl pondBOT
Available help channel!
sharp smelt
How am I supposed to interpret this
pearl pondBOT
How am I supposed to interpret this
sharp smelt
Like L(1|x_i=0)
sure,but what parameter were we even talking about given for f_i, both parameters are fixed
pearl pondBOT
@sharp smelt Has your question been resolved?
sharp smelt
.close
pearl pondBOT
pearl pondBOT
Available help channel!
crystal summit
I managed to get until this point but idk where to take this further
summer imp
Try using the product to sum formula on the LHS.
$2\sin(x)\sin(y) = \cos(x-y) - \cos(x+y)$
jolly parrotBOT
Azyrashacorki
pearl pondBOT
@crystal summit Has your question been resolved?
raven spear
Ok
crystal dew
raven spear
Ok
dark hound
when you round, the answer can get a bit messed up but its gine
raven spear
when you round, the answer can get a bit messed up but its gine
Yeah its js my teacher
dark hound
so we have 83.6, do you know hoe to find circumference
raven spear
so we have 83.6, do you know hoe to find circumference
2 Pie R
And r of 84 would be 42
dark hound
r is referribg to the radius
raven spear
r is referribg to the radius
Yea
dark hound
2 Pie R
2 pi r is the whole circumference and we only eant a portion of that, we also know the angle
so we can say we want 83.6/360 of the circumference right?
raven spear
so we can say we want 83.6/360 of the circumference right?
Oh yeah bc thats arc length formula
dark hound
yup
so just multiply 2 pi r by 83.6/360
sub in your radius and youll get the answer
raven spear
sub in your radius and youll get the answer
Wait idk if i did it wrong i got 96.4?
raven spear
calc goved me roughly 16.3 units
Hmm
Wait ik i wrote smth confusingly on my part
Im trying it agsin
dark hound
alright
raven spear
.close
pearl pondBOT
pearl pondBOT
Available help channel!
north talon
Determine all functions $f: \mathbb{N} \to \mathbb{N}$ such that for all natural $x,y$, [ f(x)^2 + f(y) | (x^2 +y)^2 ]
jolly parrotBOT
Copter
north talon
for x=y=1 we get f(1)^2 +f(1) | 4, and the only value that satisfies this is f(1) = 1
now take x = 1, y = p-1 >= 1 for a prime p, 1 + f(p-1) | p^2 so 1+f(p-1) ∈ { 1,p,p^2} but since f(p-1) >= 1 we have 1 + f(p-1) ∈ {p,p^2} for all primes p
north talon
nah
ivory basin
Cool
north talon
i think i showed injectivity, but idk if this works
wide apex
**Copter**
i think the only function which satisfies this is f(n) = n
ivory basin
Same, but the proof isn't very trivial
ivory basin
i think i showed injectivity, but idk if this works
Show
north talon
suppose that we have f(a) = f(b) and assume ftsoc that a > b then we know that
f(a)^2 + f(y) | (a^2 +y)^2 and (b^2 +y)^2 so then f(a)^2 +f(y) | gcd(a^2 + y, b^2 +y)^2 for all y, then
f(a)^2 +f(y) | gcd(a^2 -b^2 , b^2 +y)^2 for all y, but we can pick y large enough such that b^2 +y is a prime > a^2 -b^2 so they are coprime, so f(a)^2 + f(y) |1, contradiction since f(a)^2 + f(y) >= 2
not sure if this works
wide apex
we know f(1) = 1, so if we fix x=1 it becomes 1+f(y)/ (1+y)^2. Now when y=2, f(y) E {2,8} and when y=3, f(y) E {1,3,7,15}......soo in these we can only see that f(n) = n works.......
north talon
we know f(1) = 1, so if we fix x=1 it becomes 1+f(y)/ (1+y)^2. Now when y=2, f(y...
huh? why cant f(2) = 8 and f(3) = 7, or something
random ermine
suppose that we have f(a) = f(b) and assume ftsoc that a > b then we know that
...
how did you get f(a^2) +f(y) | gcd(a^2 -b^2 , b^2 +y)^2
north talon
gcd(a^2 +y,b^2 +y) = gcd(a^2 +y -(b^2 +y), b^2 +y), no?
wide apex
the original equation involves two variables, x and y which you can choose independently, so to get all values consider 2 cases : when x=1 and y=n AND when x=n and y=1. Now since f(1)=1, set x=1 and y=2 and find f(2) and then set x=2 and y=1 and find f(2). answer will be the intersection or the common function of the both cases
random ermine
gcd(a^2 +y,b^2 +y) = gcd(a^2 +y -(b^2 +y), b^2 +y), no?
ok but how did you turn the f(a)^2 into f(a^2)
north talon
the original equation involves two variables, x and y which you can choose indep...
f(2) = 2
north talon
ok but how did you turn the f(a)^2 into f(a^2)
oh ,mb typo
random ermine
ok then ur proof looks right
north talon
but the same logic still works?
random ermine
but i don't see why b^2 + y should be bigger than a^2 - b^2
north talon
works for all y, so you can choose large enough y
e.g pick y > a^2 -2b^2 to be a prime
then b^2 + y > a^2 -b^2
north talon
how would we find f(n) = n for all n though TwT
north talon
**Copter**
<@&286206848099549185>
north talon
if legendre's conjecture is true then we can use strong induction
whats that
thorny radish
there is a prime between n² and (n + 1)² for n ≥ 1
north talon
oh okay
thorny radish
that is not how induction works
north talon
yea..... if we do it for 3 we get f(x) = 3 ....... by induction, f(n) = n
2 only worked because 2^2 +1 and 1 + 2 are both prime, no?
you cant guarantee that for all n
north talon
if legendre's conjecture is true then we can use strong induction
if we strong induct then we have f(k) >= k, right? by injectivity
wide apex
f(3) is also 3......
thorny radish
if we strong induct then we have f(k) >= k, right? by injectivity
we would know f(k) = k for all k
north talon
i mean in the induction step
thorny radish
you take x := k and then let y range between 1 and k-1 to search for the value where x² + y is a prime
by strong induction we know f(y) = y
wide apex
we would know f(k) = k for all k
will this be true for all k when k belongs to natural numbers?
north talon
just prove legrendes conjecture then frfr
wide apex
it is proved that there is always a prime or a semiprime between consecutive squares.......
pearl pondBOT
@north talon Has your question been resolved?
pearl pondBOT
pearl pondBOT
Available help channel!
winged pulsar
Can anyone tell me how this formula works and why it works?
pearl pondBOT
Can anyone tell me how this formula works and why it works?
@winged pulsar Has your question been resolved?
sweet junco
Can anyone tell me how this formula works and why it works?
just to confirm this is vector projection of b on c right
winged pulsar
Yes those are finding area of triangle based on vectors
sweet junco
Yes those are finding area of triangle based on vectors
which part are you confused about
sweet junco
How is this formula found
ok so it says there on the earlier page that the area of the triangle is 1/2 * h * AB yes? you can also see it on the triangle in case it wasn't too clear
well, as in the pages, sin(v) = h/AC, we want to find only h so now AC * sin(v)=h
now, going back to the area of the triangle, substituting h=AC * sin(v), we get 1/2 * (AC * sin(v)) * AB, which looks familiar doesn't it
pearl pondBOT
@winged pulsar Has your question been resolved?
pearl pondBOT
@winged pulsar Has your question been resolved?
pearl pondBOT
pearl pondBOT
Available help channel!
alpine tusk
hi, what should i have in mind here?
pearl pondBOT
hi, what should i have in mind here?
plush bramble
hi, what should i have in mind here?
find the directional derivative first
alpine tusk
i know it has the direction of the gradient
but i dont know if i should start by more fundamental principles
also, can i just assume the function is given by 3x² + (1 - x²)?
plush bramble
that's probably fine too. then you don't need directional derivative formula at all and just maximize a single variable function
alpine tusk
oh but i think it would not be the way the autor wants us to solve the problem
plush bramble
great do the way i said then
alpine tusk
i did and it is (6x, 2y) but i dont know what to do next
maybe letting a = 6kx and b = 2ky since the rate of change is maximum on the direction of the gradient
plush bramble
the "value" of a directional derivative vector is simply the norm/magnitude of the vector
left anchor
Any idea how I could rewrite e^(npi(i^n))
pearl pondBOT
Any idea how I could rewrite e^(n*pi*(i^n))
left anchor
It’s a sequence that I’m supposed to examine to figure out whether it’s convergent or not
But I am not sure how to do that in its current form
hoary relic
Any idea how I could rewrite e^(n*pi*(i^n))
$e^{n \pi (i^n)}$
jolly parrotBOT
multiplexer
hoary relic
This?
left anchor
Yes that
rough forge
Maybe you can split it into 4 cases since i repeats for every multiple of 4
left anchor
like pi, 2pi and the same thing *(-1)?
rough forge
no like, for k>=0
i^0=1 -> 4k
i^1=i -> 4k+1
i^2=-1 -> 4k+2
i^3=-i -> 4k+3
...
fast radish
idk if allowed to ask but can someone help me with preperation for my math exam tomorrow? just help me learn a bit from my mistakes i did on some tests, and maybe some tips for the exam. (9th grade)
rough forge
idk if allowed to ask but can someone help me with preperation for my math exam ...
this channel is occupied, see #❓how-to-get-help
fast radish
thanks
rough forge
**multiplexer**
If you consider the subsequence, where n=4k it shouldring a bell what happens with n->oo
left anchor
no like, for k>=0
*i*^0=1 -> 4k
*i*^1=*i* -> 4k+1
*i*^2=-1 -> 4k+2
*i*^3=-*i* ->...
can i not split it into sth like sin + cos * i and just look at the result?
or does it not work that way
left anchor
i assume not
we can do that too
do i just write it down for the first few n and see what i get
left anchor
is there a way to tell that 4k will be the biggest result?
could it not be k3 for example?
rough forge
from the 4 cases we take a look for n=4k
left anchor
why 4k specifically was my question
rough forge
you will find out
what happens to your expression
if n=4k what is i^n=? for all n?
rough forge
yes so what do you have left
left anchor
yes
rough forge
How does exp(npi) behave as n->oo
left anchor
it do oo
rough forge
yes, so we found one subsequence that doesn't converge
left anchor
oh right
i didnt think that far
so we know its impossible for the entire thing to converge
rough forge
yes
left anchor
and we can examine the rest to figure out accumulation points
(unfortunately i need to do that too)
rough forge
yes
rough forge
you could do as a piecewise sequence
jolly parrotBOT
left anchor
Does this work
rough forge
i think you messed up
left anchor
I did
rough forge
the second is with i since 4k+1 = 1 mod 4
the third is -1 since 4k+2 = 2 mod 4
like verything is off
rough forge
small example i²=-1 but so is i² times i⁴ = i⁶ = -1
because i⁴=i² times i²=(-1)(-1)=1
the mod is just there to make sense of the higher offsets
pretty much 4k+1 is the same as just 1
left anchor
I changed it to this earlier but my brain broke for good
Can’t tell if it makes more or less sense now
rough forge
I changed it to this earlier but my brain broke for good
yes
imagine like, for evertime you just multiply another i
from 1 -> 2 you multiply i by itself so -1
from 2 -> 3 you multiply -1 with i so -i
and from 3 -> 4 you multiply -i with i so 1
left anchor
Is i defined to be positive or negative
left anchor
imagine like, for evertime you just multiply another *i*
Yeah i tried slowly doing that earlier when I had to first figure stuff out
left anchor
Is i defined to be positive or negative
Idrk what to with exponents like n*pi * i
rough forge
Is i defined to be positive or negative
the thing with complex numbers is we can't really compare them in the first place
left anchor
Oh so they don’t have any relation?
left anchor
If my elements are all complex
How do I know whether it converges or not
If convergence relies on that relation
rough forge
well we use modulus
z_n -> L if |z_n-L| -> 0 or < eps basically
or z_n=x_n+iy_n converges iff real and imaginary parts converge
left anchor
or z_n=x_n+*i*y_n converges iff real and imaginary parts converge
I knew that thankfully
But how do I examine the imaginary part now
Do I just look at the coefficient of i?
rough forge
well for that you can actually do what u wanted to do earlier
left anchor
Trig functions?
rough forge
yeah
but i don't think it's necessary, you can directly take the limit of z_n
you will just get a complex number
rough forge
I changed it to this earlier but my brain broke for good
like the 2nd one behaves like a real sequence, where as first and third are periodic
jolly parrotBOT
left anchor
Does the second case approach 0?
rough forge
Does the second case approach 0?
Ja
jolly parrotBOT
left anchor
I hope I finally understood what you mean
rough forge
It's like a clock
After a day a cycle passed (24h) so we restart at 0 not keep going up
left anchor
I was trying to understand why we are left with pi*i
And not just i
But I think that’s because we have an odd coefficient
And we’re doing mod 2pi
Like 5pi*i mod 2pi
Right
rough forge
you can't subtract 2pi multiples off ipi
rough forge
ok yeah i should have only written the coefficient since we are interested in the angle
jolly parrotBOT
left anchor
I don’t fully understand the last line
Does that mean we can just rewrite it as e^pi*i
Because of the mod
left anchor
Yeah but the entire exponent is npii
What did we do with the i
rough forge
nothing
we just wanted to simplify the angle
to make more sense of the expression
if you want, you can also write some values down for k
maybe it's easier to understand in trig form
cos(theta)+isin(theta)
cos((4k+1)pi)+isin((4k+1)pi)
cos(2pi(2k)+pi)+isin(2pi(2k)+pi)
cos(x) and sin(x) are 2pi-periodic, idk what to tell you
so you can go subtract 2pi then 2k many times, leaving you with pi
left anchor
Sorry I’m genuinely not in the best state rn
I think I got the point
rough forge
Ah oka
jolly parrotBOT
rough forge
hope das macht es verständlicher, gtg
left anchor
Should be fine thank you
frigid spindle
Where the hell did I fuck up? I keep having trouble with the algebraic part when doing partial fractions
frigid spindle
My answer is the one of the bottom left and the correct one is on the bottoms right from the textbook
plush bramble
how did you get this eqn
frigid spindle
Well I figured (x+1)^2 would be split into two part and then each of them is multiplied by the other's denom
plush bramble
yes that's the first part
plush bramble
how did you get this eqn
that doesn't explain how you arrived here
plush bramble
the bottom one with k=2
plush bramble
how did you get this eqn
whatever you did between the screenshot i sent and here was wrong
but you skipped steps so i can't point it out to you
frigid spindle
What??? I had no idea I was skipping steps
plush bramble
from here
plush bramble
how did you get this eqn
what did you do to get here
frigid spindle
whatever you did between the screenshot i sent and here was wrong
I multiplied A with (x+1)^2 and B with (x+1)
plush bramble
Why would you do that
frigid spindle
what did you do to get here
Factored the x^2+2x+1 in the denominator in the original integral
plush bramble
yea that part was fine, it was the previous step.
plush bramble
I multiplied A with (x+1)^2 and B with (x+1)
still doesn't explain why you'd multiply A by (x+1)^2 out of nowhere
frigid spindle
Then what am I supposed to multiply by?
plush bramble
higher's gonna spoil it isn't he
blazing garden
higher's gonna spoil it isn't he
Does he always give away answers rather than guiding people?
plush bramble
$\frac{2x + 3}{(x+1)^2} = \frac{A}{x+1} + \frac{B}{(x+1)^2}$
jolly parrotBOT
riemann
frigid spindle
Ok?
plush bramble
do you know where this comes from?
frigid spindle
Yes
plush bramble
do you know what the common denominator is?
frigid spindle
It's 1?
plush bramble
(x + 1) and (x+1)^2 share a common factor
frigid spindle
Or well x+1 more specifically
plush bramble
yes do you know how to put the right side as a single fraction
prime bramble
higher's gonna spoil it isn't he
no, I was gonna ask something, but then found that I was being silly 
frigid spindle
yes do you know how to put the right side as a single fraction
Like do the addition?
plush bramble
sure
plush bramble
no
frigid spindle
Then what
plush bramble
**riemann**
the point is to match the left side
you increased your denominator's power for no reason
plush bramble
do you know what the common denominator is?
maybe i should have said least common denominator here
verbal whale
It's like doing $\frac12 + \frac14 = \frac48+\frac28$ ...
jolly parrotBOT
Alberto Z.
plush bramble
I did the addition
but not with the least common denominator
verbal whale
**Alberto Z.**
@frigid spindle
frigid spindle
**Alberto Z.**
I think that's what I did?
verbal whale
**Alberto Z.**
I hope you wouldn't do this , right?
verbal whale
I think that's what I did?
Yes, that's the problem
frigid spindle
but not with the _least_ common denominator
Least common denom would be 1 ith
plush bramble
no
frigid spindle
Wait wait
plush bramble
**riemann**
what do you multiply x+1 by to get (x+1)^2 ?
verbal whale
$\frac1x + \frac{1}{x^2} = ?$
jolly parrotBOT
Alberto Z.
frigid spindle
what do you multiply x+1 by to get (x+1)^2 ?
By x+1
plush bramble
yea so do that instead of what you did here
verbal whale
**Alberto Z.**
What would the least common denominator be here?
It's not x³, do you agree?
frigid spindle
Yes
verbal whale
Good, so do you now see the "issue" in what you did?
pearl pondBOT
pearl pondBOT
Available help channel!
echo nebula
how to take out the range of values of k where f(x) = k only has one solution
jolly parrotBOT
Civil Service Pigeon
pearl pondBOT
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
honest oyster
Multiply x^3 on both sides: 5x+1=kx^3 has only 1 sol
cursive wolf
You probably don't want to try and solve the equation for x or so, that may be annoying with cubics. But if you only care about finding uniqueness, it's enough to understand what the graph looks like
i.e. find local maxima and minima, and from that deduce where the function assumes which values
echo nebula
i found the maxima
it was 18.52
however many values under the maxima did have 2 or 3 solutions
so i was confused
cursive wolf
Multiply x^3 on both sides: 5x+1=kx^3 has only 1 sol
Rewriting the equation like this can make it a bit more intuitive
Because then you're really just looking for unique values of a cubic polynomial, and you probably have an idea of what cubics look like
echo nebula
Because then you're really just looking for unique values of a cubic polynomial,...
uh, btw
graphing display calculato
was allowed for this q
hasty saddle
if graphing calcs are allowed, just graph f(x) and look at it
smoky gull
graphing display calculato
whats the prob then
kindred frost
graphing display calculato
Then just graph the function
pearl pondBOT
@echo nebula Has your question been resolved?
rigid terrace
I have to turn [ ( x^3 ) / ( x-3 ) ] into a power series. I think I have it, but I'd very much appreciate having my work checked.
dense jasper
I have to turn [ ( x^3 ) / ( x-3 ) ] into a power series. I think I have it, but...
,w sum of -(1/3)^(n+1) x^(n+3) from n=0 to inf
jolly parrotBOT
pearl pondBOT
dense jasper
oh, thank you! sorry to take up space here
pearl pondBOT
Available help channel!
daring steeple
hello?
daring steeple
is anyone online?
past perch
no
maiden crest
Please only use these channels to ask questions 
pearl pondBOT
Asking the actual question right away is more likely to get responses.
Asking "Can I ask...?" or "Does anyone know about...?" doesn't give people enough information to decide whether they can help, and answering can feel like a promise to help with the actual question, which they might find themselves unable to.
maiden crest
Ok
maiden crest
Do you know how to differentiate/integrate sin and cos?
daring steeple
Nah it aint explain in school yet
maiden crest
Alright
ebon skiff
!status
pearl pondBOT
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
can you explain me?
pearl pondBOT
@daring steeple Has your question been resolved?
sharp smelt
pearl pondBOT
sharp smelt
Hi, I don't really get which test I'm to use here
Which of the Z tests that is
" standard error of difference between the sample mean" is confusing me
lament dawn
Which of the Z tests that is
retroactive oops (but yes two sample z test is right)
sharp smelt
Like I get I'd want to use this
lament dawn
mhm
sharp smelt
So to determine if this error is significant I'd find the z score for a two tail test right
lament dawn
yep
sharp smelt
okay and here I take a 99% confidence right
the Z score is 2.741
at a 0.01 level of signifiance
2.575≤2.741
so we reject the null hypothesis
lament dawn
yep (im not sure if you z score is correct though is your SE 1.032?)
sharp smelt
yep (im not sure if you z score is correct though is your SE 1.032?)
lemme do that, I forgot to calculaute that
lament dawn
lemme do that, I forgot to calculaute that
but anyways you would get a number that would be greater than 2.575 and by extension reject the null
lament dawn
if youve no more queries yep
sharp smelt
.close
pearl pondBOT
pearl pondBOT
Available help channel!
earnest warren
For any natural number $n>2$, show that:
[
\sum_{i=1}^n ( \alpha_i \cos(ix)) = \sum_{i=1}^n (\beta_i \sin(ix)) \hspace{1cm} \forall x \in \mathbb{R}
]
If and only if $\alpha_1=\alpha_2=\dots =\beta_1 =\beta_2=\dots = 0$
jolly parrotBOT
Alexis_Fx
earnest warren
Smth with induction idk
vital crescent
n>2 feels weird to me
earnest warren
Original problem
vital crescent
what have u done so far
earnest warren
Honestly, nothing. this's the only proof-based problem on the sheet and I'm kinda bad at it
dusty jungle
maybe euler's formula can help
earnest warren
Oh wait....
vital crescent
Honestly, nothing. this's the only proof-based problem on the sheet and I'm kind...
sure but i think that might just be that the question is hard, writing the proof might be another thing we could talk abt but right now we just wanna solve the q
earnest warren
maybe euler's formula can help
Wait hang on
dusty jungle
im hanging
earnest warren
This's an old paper from 2017, we used to have complex number in our syllabus
Used to
As in complex number is no longer in out syllabus since 2018
earnest warren
I barely know what euler's formulas is, Imma take a look at it, thx
vital crescent
euler's formula would make it easy
it does?
dusty jungle
it does?
it transforms it into a question about polynomials
but clearly its not required here
vital crescent
it transforms it into a question about polynomials
how so?
I dont immediately see what u mean
considerin this is not the intended solution we might as well talk about it with spoiler
dusty jungle
how so?
||basically the equation will become a sum of (e^ix)^k where k is some power and with some coefficient. so this hold for all x. if we let e^ix = y, the equation is a polynomial in y, and it has infinite roots(in the unit circle), so its the zero polynomial so all coefficients are 0||
vital crescent
||basically the equation will become a sum of (e^ix)^k where k is some power and...
ah right