Which is what I sent in the photo
#help-26
neon iron
topaz sinewBOT
@neon iron Has your question been resolved?
sudden dagger
topaz sinewBOT
sudden dagger
I have to check to see if (A+B)(A-B) is symmetric if A = A' and B = B'
The answer booklet says it isn't symmetric
But my working came to the conclusion that it is symmetric
(A+B)(A-B) multiplied out is not A^2 - B^2 in either case. because multiplication isn't commutative, so we should get AA - AB + BA - BB
loud oasis
as you said, matrix multiplication is not commutative
in one case you get -AB + BA and in the other you get -BA + AB
sudden dagger
But addition isn't order-dependent is it?
wait
Ohh
I see
minus BA for the second one, and - AB for the first one
they are different, mb
thank yoiu
.close
topaz sinewBOT
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hoary fable
part b
sharp chasm
topaz sinewBOT
part b
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
@hoary fable Has your question been resolved?
twilit kindle
Is this done correctly?
topaz sinewBOT
Is this done correctly?
ember lodge
umm
you first minus eight at both sides
then how come the right side still 20
it should be equal to 12 since 20 - 8
thorny flameBOT
jandro
ember lodge
ye u did right
twilit kindle
my teacher did that work
twilit kindle
I see, was a bit confused there
ember lodge
oh, lol
twilit kindle
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topaz sinewBOT
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visual spindle
I need help regarding this task: Let S and T be two independent geometrically distributed random variables. S has parameter p and T has parameter q with 0 < p, q ≤ 1. Show that U= min{S,T} is geometrically distributed by determining P(U > n). What is the parameter of the geometric distribution of U?
My idea was that P(U > n) = P(S > n) * P(T > n) . After that, I rearranged this term using the geometric series as shown in the picture. But I'm not sure if that's enough to prove that U is geometrically distributed.
topaz sinewBOT
I need help regarding this task: Let S and T be two independent geometrically di...
@visual spindle Has your question been resolved?
topaz sinewBOT
@visual spindle Has your question been resolved?
dense rain
I would expect that using MGFs is the easiest method here
visual spindle
Update
could you maybe explain how to use MGF? its the first time I hear of that😅
dense rain
Have you learned what a moment generating function is?
If you haven't then that won't really help you
visual spindle
not in my probability course, maybe if its also part of linear algebra
dense rain
You will probably learn it soon, but that won't help you now
visual spindle
Aight
idk my updated term has at least kind of the form of a geometrical distribution, do you think I could work with that?
dense rain
I misread your problem and didn't see the min(S,T) part and can't recall the order statistic formula off the top of my head.
Your immediate problem is that you applied the geometric series formula incorrectly here
dense rain
I need help regarding this task: Let S and T be two independent geometrically di...
.
Your starting index is n so you would need to account for the the missing terms, I also haven't looked if what you wrote is a reasonable distribution for the minimum, but it seems okay at a glance.
visual spindle
its the first time that I'm really dealing with geometric series, so I guess I#d probably have something like the geometric series minus a sum starting from 0 to n right?
dense rain
yeah
I don't know how that actually helps you, it just kind of looks like it makes things more complicated. It'd be easier to see when it's all written down
visual spindle
nice, imma check that!
i have used the partial sum for the geometric series so resolve the second sum
i need to review it again i messed up some stuff
dense rain
I think you can get there doing what you are doing, but looking at it again, you are over complicating this by doing all the explicit calculations. For a probability course, they usually expect you to short cut some of these steps once the formulas are given to you because this is a probability course, not a calculus course.
You already know that the CDF of S and T is 1-(1-p)^k and 1-(1-q)^k respectively. So P(S>n) = 1 - P(S < n) = (1-p)^k for example. You really don't need to do it all directly from first principles although that is good practice for other math courses.
I should write <= there, but I'm not going to edit it
topaz sinewBOT
visual spindle
.reopen
topaz sinewBOT
✅
visual spindle
So using CDF my calculations result in this term but I’m still not sure how this helps me showing that U is geometrical distributed
i guess I should reuslt in something like this
so basically I'm missing my parameter ouside of the brackets but I dont know where I'm loosing it
jagged snow
topaz sinewBOT
jagged snow
Is the second limit equal to 3?
Since $\lim_{x \to 1^+} f(f(x)) = f(\lim_{x \to 1^+}f(x)) = f(3) = 3$
lean rock
Yeah
thorny flameBOT
WhoTao
topaz sinewBOT
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ocean lion
<@&286206848099549185> when is arctanx>0
topaz sinewBOT
<@&286206848099549185> when is arctanx>0
meager dawn
thorny flameBOT
ocean lion
,w arctanx>1
thorny flameBOT
topaz sinewBOT
@ocean lion Has your question been resolved?
signal path
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final wren
If I have some integral, can I always make a function v(t) such that x = v(t) and then the integral on f(x)dx is the same as f(v(t))*v'(t)dt?
topaz sinewBOT
If I have some integral, can I always make a function v(t) such that x = v(t) an...
worldly osprey
I mean, v(t)=t will work
final wren
wdym
worldly osprey
If v(t)=t, then f(v(t))v'(t)dt=f(t)dt, and the integral of f(t)dt is the same as the integral of f(x)dx. Not a very interesting answer, but as your question is written, the answer is "yes" for this reason
final wren
ok so then if I have any integral
$\int :f\left(t\right)dt:=:\int :f\left(v\left(t\right)\right)v'\left(t\right)dt$ This is true?
thorny flameBOT
sloppymope
worldly osprey
Only if v(t)=t
final wren
oh
worldly osprey
In general, maybe not
final wren
right
Then is this true:
$\int :f\left(x\right)dx=g\left(x\right),:x=v\left(t\right):then:\int :f\left(v\left(t\right)\right)v'\left(t\right)dt:=:g\left(t\right)$
thorny flameBOT
sloppymope
final wren
g(v(t))*
worldly osprey
g(v(t))*
Yes, I believe so. There may be some domain considerations (like if v(t) has a singularity), but otherwise I think that works
worldly osprey
If the right side is true then so is the left
I'm pretty sure that this direction is just the classic calc 2 u-substitution, written differently
final wren
Its like the inverse of u sub
worldly osprey
Maybe not "inverse" per se, but yeah. I've seen both directions called u sub before
final wren
For this "inverse" does v have to be inverible?
worldly osprey
Yeah I suppose it does, depending on how strict you plan on being with domains
final wren
Why does it have to be?
worldly osprey
Maybe not actually. I'm second-guessing myself rn
final wren
I am quite sure it does, I just dont really understand why
worldly osprey
**sloppymope**
In the context of a definite integral (from a to b, for example), it does if you're doing this first direction, since you'd need to be able to solve a=v(t) and b=v(t). But I'm pretty sure there are still ways around that if v isn't globally invertible
For the reverse direction, being able to solve x=v(t) for t may make it easier to substitute it in the integrand, but may not be necessary if the integrand is set up in a nice enough way? I'm having a hard time remembering some of these integration properties; it's been a while
topaz sinewBOT
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gusty cobalt
can someone help me with this question
topaz sinewBOT
can someone help me with this question
gusty cobalt
i dont get it
loud oasis
do you know the impulse momentum theorem?
loud oasis
so find the momentum before, and use the impulse to find the momentum after
topaz sinewBOT
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carmine wren
Hello i have a question and would appreciate your help.
Construct a mathematical model of the problem and solve the simplex by the method:
For construction, you need sets of boards, each of which consists of 4 boards of length 1.5 m and 2 boards of length 2 m, 1 board of length 2.5 m. How should 500 5-meter boards be sawn to get the maximum number of sets.
I have not idea how construct a mathematical model of the problem.
topaz sinewBOT
Hello i have a question and would appreciate your help.
Construct a mathematical...
trim bay
can someone help me
trim bay
my friend told me that the formula for r is p=m*v
carmine wren
maybe you know my 1
Construct a mathematical model of the problem and solve the simplex by the method:
For construction, you need sets of boards, each of which consists of 4 boards of length 1.5 m and 2 boards of length 2 m, 1 board of length 2.5 m. How should 500 5-meter boards be sawn to get the maximum number of sets.
carmine wren
i dont know how Construct a mathematical model of the problem
carmine wren
Hello i have a question and would appreciate your help.
Construct a mathematical model of the problem and solve by the simplex method:
For construction, you need sets of boards, each of which consists of 4 boards of length 1.5 m and 2 boards of length 2 m, 1 board of length 2.5 m. How should 500 5-meter boards be sawn to get the maximum number of sets.
I have not idea how construct a mathematical model of the problem.
<@&286206848099549185>
topaz sinewBOT
@carmine wren Has your question been resolved?
carmine wren
<@&286206848099549185>
neon iron
what
carmine wren
For construction, you need sets of boards, each of which consists of 4 boards of length 1.5 m and 2 boards of length 2 m, 1 board of length 2.5 m. How should 500 5-meter boards be sawn to get the maximum number of sets.
i need solve that
using simplex method
I don't know how to find the constraints and objective function
carmine wren
.close
topaz sinewBOT
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neat talon
Hi
topaz sinewBOT
@neat talon Has your question been resolved?
neat talon
Not yet
neat talon
<@&286206848099549185>
wheat kestrel
neat talon
Hi
Finally someone
I have a nightmare question I’ve been trying to solve for couple hours
I hope you can help me
neat talon
Ive been trying to use a rational functions, Ive used f(x)= - 2x/4x - x/2x -1 to get an instantaneous rate of change of 1.23
But i get 0
neat talon
Ok
neat talon
Are you there?
neat talon
<@&286206848099549185>
eternal olive
Yo
austere musk
i manged to get everything except the last criteria :/
oh wait
nvm i think i got it
nvm i dont
xD
yeah @neat talon ive been tryna figure it out for a bit but cant
i managed to get every criteria except the last one where they intersect
y = (9.58x - 2) / (x + 3)
y = -2.96sin(x)
see if you can play around with these and get them to intersect at x = 2
without messing up the other criterias
because they both have instaneous rate of change of 1.23 at x = 2
neat talon
yeah <@1225644738166788179> ive been tryna figure it out for a bit but cant
Ow
Thanks a lot
neat talon
because they both have instaneous rate of change of 1.23 at x = 2
I’ll try to
How did you get them?
austere musk
basically just setting up constants and working backwards to solve for them
y = (Ax-2)/(x+3)
take the derivative, solve for A at x = 2
do the same for the other trig function
y = Bsin(x)
take derivative and solve for B at x = 2
but the final criteria is still missing
you need to set those functions equal to each other and solve for some similar constant
but then you would have two constants
which is why i couldnt figure it out
like u need the same number of equations as missing variables
the derivative ends up changing and breaking the criteria if you start multiplying with more constants
i cant figure out but someone might or if u figure it out let me know
im interested what pair of functions satisfies that criteria
neat talon
the derivative ends up changing and breaking the criteria if you start multiplyi...
Ok I see
You’ve solved literally 90 % of the question , I’ll try figuring out the rest and i’ll let you know
Thanks a lot ni
austere musk
np good luck!
neat talon
How did you get the instantaneous rate of change at 1.23 @austere musk
austere musk
take the derivative, solve for A at x = 2
^
its instantaneous rate of change at x = 2
which is equal to 1.23 (rounded)
neat talon
Ive tried with these pair if equation but i got 10 with the trigonometric one
Ive tried f(2+0.0001)-f(2)/0.0001
austere musk
not sure what f is
neat talon
If means the whole function
austere musk
thats not instantaneous rate of change, thats a really really really small rate of change
neat talon
Ow
austere musk
instantaneous rate of change = derivative
neat talon
I don’t think i read it in my mhf4u course
austere musk
not sure what mhf4u stands for
neat talon
They only teached us one formula
neat talon
not sure what mhf4u stands for
Advanced function
austere musk
if they didnt teach you derivatives
then idk how they expect u to solve this equation
neat talon
They only teached us one formula
For the instantaneous rate of change
neat talon
then idk how they expect u to solve this equation
I hate ilc school
@austere musk
I found it
F(x) = (9.58x-6)/(x+3)
And g(x)= 12 sin(x) - 2
neat talon
They both intersect at y=-2
neat talon
that doesnt intersect the y axis
I dont get it
austere musk
dang ur right
neat talon
Really?
neat talon
How can i check the instantaneous rate of change?
austere musk
How can i check the instantaneous rate of change?
take the derivative and plug in 2
neat talon
is 12cos(2) = 1.23?
-4.99
neat talon
take the derivative and plug in 2
Ok
neat talon
Can i use log to find out?
topaz sinewBOT
@neat talon Has your question been resolved?
topaz sinewBOT
@neat talon Has your question been resolved?
topaz sinewBOT
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fading venture
hello :) My teacher wants me to explain the prolate spheroid's radius, but I don't know how. Can someone pls help me, appreciate it :)
topaz sinewBOT
hello :) My teacher wants me to explain the prolate spheroid's radius, but I don...
topaz sinewBOT
@fading venture Has your question been resolved?
neon iron
The variables (x) and (y) are obtained by the equation (py^2+\frac{qy}{p}=x). Diagram shows the straight line graph obtained by plotting y againts (\frac{x}{y}).
topaz sinewBOT
The variables \(x\) and \(y\) are obtained by the equation \(py^2+\frac{qy}{p}=x...
thorny flameBOT
$ flashys
topaz sinewBOT
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somber venture
topaz sinewBOT
somber venture
hey can i confirm that the answer is all good?
bc when i use mathway to check it gives me a general solution… just wondering if that’s necessary for this question?
charred rampart
hey can i confirm that the answer is all good?
domain
charred rampart
think back to the unit circle
somber venture
yes
somber venture
yeah
charred rampart
the domain is in the negatives
somber venture
oh oh
somber venture
and 2pi
i’m kinda confused still
wait so
am i just
like
choosing a random value from the unit circle
to subtract
and do i subtract from pi/6 or the negative pi/6??
topaz sinewBOT
@somber venture Has your question been resolved?
somber venture
<@&286206848099549185>
topaz sinewBOT
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final swift
How do I solve this? The question is Prove by induction that n^3 + 3n^2 + 8n is divisible by six
topaz sinewBOT
How do I solve this? The question is Prove by induction that n^3 + 3n^2 + 8n is ...
dusk apex
First step of induction is show that it is true for the first term that you want to prove it for
dusk apex
Let's say that P(n) is the statement that n^3 + 3n^2 + 8n is divisible by 6
then P(1) states that 1+3+8 is divisible by 6, meaning that 12 is divisible by 6, which is true
final swift
I get all of it but I cannot prove its divisible by six once I sub in k+1
vernal shard
I get all of it but I cannot prove its divisible by six once I sub in k+1
Show your working so far
final swift
Ye lemme write it cause it was on a test I just did
dusk apex
then you use P(k) to show that P(k+1) is divisible by 6
P(k+1)-P(k)
this is obviously divisible by 6
final swift
Hoooww
final swift
Bruuuuuh thats soo dumb I thought I was supposed to get a 6 outside the bracket
final swift
Trust me if there was a way I woulda found it I wrote like 2 pages on it
Cause I kept getting that but didnt know it was right
But wait lemme show working out
dusk apex
nah you cant get a 6 outside of a bracket
what i just did is mathematically valid though
and it is induction
final swift
Hopefully its understandable
So i can just say that both 6m and 3(3k.....) are divisible by 6?
plush tulip
are you convinced by that?
dusk apex
how i would write it:
P(k): .........., let this equal 6g, g is an element of positive integers
P(k+1): ............
P(k+1)-P(k)= ............ which is divisible by 6, let this equal 6c, c is an element of positive integers
P(k+1)-6g=6c
P(k+1)=6(g+c) which is obviously divisible by 6
final swift
Bro thats so simplified why are we made to do all this working out
dusk apex
5 lines is pretty long
plush tulip
Bro thats so simplified why are we made to do all this working out
i mean your working out is still necessary
and essentially the same
final swift
I wrote a lot more than 5 for sure
But tbf If I didnt get what was going on in the background what you wrote wouldnt make sense
plush tulip
So i can just say that both 6m and 3(3k.....) are divisible by 6?
so 6m is obviously divisible by 6 right?
final swift
Not in my test haha
plush tulip
everything is 1 line if your paper is big enough
final swift
Well ye I just didnt know that 3(3k+4+k^2) was divisible by 6
plush tulip
right
shut obsidian
So i can just say that both 6m and 3(3k.....) are divisible by 6?
Do you know some modular arithmetic?
plush tulip
we just need to show 3k+4+k^2 is even
4 is already even
so just need to show 3k + k^2 is even
final swift
Do you know some modular arithmetic?
Is it like that stuff with the remainder?
thorny flameBOT
Kepe
final swift
aah its crazy how much some of these topics link together I just know them but cant connect them unless told it can be done
final swift
Yep
shut obsidian
ok, 3k becomes k, right?
thorny flameBOT
Kepe
thorny flameBOT
Kepe
final swift
should I send?
The more the better
void yew
more of a solution thats why asking
final swift
So then u are left with k+k which is just a multiple of 2?
shut obsidian
Yep
shut obsidian
So then u are left with k+k which is just a multiple of 2?
So we've shown that the expression is 0 mod 2
Thus we are done, it's even
final swift
more of a solution thats why asking
I dont get how I would even manage to think about this
shut obsidian
I dont get how I would even manage to think about this
The mod stuff?
final swift
Like it just randomply pops up in ur mind to do that or what
shut obsidian
Well modular arithmetic is about remainders, exactly what we want here. We want to show it has a remainder of 0 when dividing by 2
It's basically the go-to method in this case
void yew
f(n)=n³ + 3n²+ 8n
f(0) = (0)³ + 3(0)²+ 8(0)
f(1) = (1)³ + 3(1)²+ 8(1)
...
f(k) = (k)³ + 3(k)²+ 8(k)
f(k+1) = (k+1)³ + 3(k+1)²+ 8(k+1)
f(k+1) = k³ + 1 + 3k² + 3k + 3k² + 6k + 3 + 8k + 8
= k³ + 6k² + 17k + 12
= k³ + 6k² + 18k - k + 12
= k³ - k + 6k² + 18k + 12
= k³ - k + 6(k² + 3k + 2) (k³-k) Has to be a multiple of 6
k³ - k has a neat property
1 = 1
4 = 1 + 3
9 = 1 + 3 + 5
...
n² = 1 + 3 + 5 + ... + (2n-1)
1 = 1
8 = 3 + 5
27 = 7 + 9 + 11
...
n³ =
shut obsidian
Like it just randomply pops up in ur mind to do that or what
It's not the only option though
You could as well prove it by another induction, for example
final swift
Im just salty I didnt get full marks cause of this question
Its so easy once dissected
shut obsidian
f(n)=n³ + 3n²+ 8n
f(0) = (0)³ + 3(0)²+ 8(0)
f(1) = (1)³ + 3(1)²+ 8(1)...
Is that complete..?
final swift
f(n)=n³ + 3n²+ 8n
f(0) = (0)³ + 3(0)²+ 8(0)
f(1) = (1)³ + 3(1)²+ 8(1)...
Aah so that -k cancels out the k^3?
That working out looks exactly like what I had just with the ending bit
shut obsidian
f(n)=n³ + 3n²+ 8n
f(0) = (0)³ + 3(0)²+ 8(0)
f(1) = (1)³ + 3(1)²+ 8(1)...
That's a nice one
final swift
Neat property?
final swift
That is pretty nice I saved it for later if I need to remember anything else before I close?
shut obsidian
That is pretty nice I saved it for later if I need to remember anything else bef...
I have one more solution that's also a neat one. Do you see how the one from Sad Poyo works?
final swift
Ye it annoys me I didnt think of that
void yew
I have one more solution that's also a neat one. Do you see how the one from Sad...
I forgot the cube series
shut obsidian
So that will be divisible by 3
shut obsidian
Wait
final swift
Aah one must be even and one must be a multiple of 3 so its a multiple of 6
Oh wait that doesnt work
shut obsidian
Let me propose one more solution directly from your working out
So, we need to show 9k + 3k^2 = 3(3k + k^2) is even.
So we need to show 3k + k^2 is even
void yew
Though that's only 3
It works
shut obsidian
that's k(3 + k)
void yew
k * (k-1) * (k-2) gives a multiple of both 6 and 3
shut obsidian
Now this will always be even, because if k is odd, then k + 3 must be even
(if k is even, we are done anyways)
shut obsidian
k * (k-1) * (k-2) gives a multiple of both 6 and 3
Oh, true
void yew
either k and k-2 are even and k-1 odd, or vice versa
either case it gives a multiple of 6
shut obsidian
k(k-1) makes it a multiple of 2
k(k-1)(k-2) makes it a multiple of 3
So in total it's a multiple of 2*3 = 6
That also works, yep
final swift
So many options
void yew
there is another proof i was going for
final swift
Yet none came to mind
void yew
like sum of odd numbers give squares
shut obsidian
I would actually prefer these over the modular arithmetic one
void yew
what sum of seires gives cubes?
final swift
When did u guys learn this ? Like dont u forget this easier stuff a bit later on when in uni?
void yew
When did u guys learn this ? Like dont u forget this easier stuff a bit later on...
just some fiddling with algebra
shut obsidian
You often need some little tricks for solving problems when studying math at uni, so you kinda get used to it too
void yew
what sum of seires gives cubes?
@shut obsidian
there's another graphical proof i was going for
cant remember the sequence
shut obsidian
<@332561465405865985>
Uh, $n^3 = \sum_{k=0}^n ((k + 1)^3 - k^3)$?
thorny flameBOT
Kepe
final swift
I'm struggling to find a topic that I find fun to do a harder problem on it for my personal statement you think there is questions like this maybe harder or does it just get insanely hard after this point
shut obsidian
Probably you mean something else
void yew
like the sum of odd numbers gives squares
1 + 3 + 5 + ... + n = n^2
There's another proof
lemme check the Pascal's triangle
final swift
Are you just competing to see who can get the most proofs or smth?
dull sonnet
1 + 3 + 5 + ... + n = n^2
There's another proof
you mean, 1 + 3 + 5 + .. + (2n-1) = n^2
void yew
you mean, 1 + 3 + 5 + .. + (2n-1) = n^2
thnx for correction
also found it
it was also sum of odd but with each skip it gives you next power of 3
shut obsidian
I'm struggling to find a topic that I find fun to do a harder problem on it for ...
Yeah, of course there are
Do you mean specifically stuff on induction?
final swift
Like the problem I had in my statement was sooo basic and I didnt even make it well so I might switch it to some induction or proof
void yew
Like the problem I had in my statement was sooo basic and I didnt even make it w...
wanna know the best part, I studied induction methods on my 2nd semester, like 4 years ago, and forgot everything
final swift
wanna know the best part, I studied induction methods on my 2nd semester, like 4...
Haha that is with every topic tbh
shut obsidian
Like the problem I had in my statement was sooo basic and I didnt even make it w...
I can give you some induction problems that were given during the first weeks of uni
final swift
That would be really good
void yew
I can give you some induction problems that were given during the first weeks of...
same pls, i wanna practice more
ik it wont help me in my job but still fun to do induction
final swift
Like let me show you what weak problem I made up to woffle about in my statement
Another aspect I like about math is that it is fun to apply the concept of probability to daily occurrences. For example, I get the bus every day to college and I've noticed that about once a month the bus comes earlier than usual, using this information I used Poisson distribution to work out my probability of me missing the bus 2 days in a row given I arrive to the stop at the set time it is supposed to arrive. I used this as one day as I had missed the bus 2 days in a row and I wanted to prove to my family how unlikely it was for this to happen. So I used λ=21 to signify how many times a month the pass comes by the stop and tested for x=2 and my answer was 1.672x 10^(-7).
void yew
ffs just found a simpler version of my solution
final swift
I thought I was cooking until I realized I was writing this at 5am yesterday with no sleep
void yew
k^3 + 3k^2 + 8k
k^3 + 3k^2 + 9k - k
k^3 -k + 3k^2 + 9k
(k-1)(k)(k+1) + 3k(k + 3)
Both sides multiple by 6
4 line solution
final swift
Bro it was literally a 6 or 8 mark question for me
shut obsidian
\begin{problem} Prove the following using induction: \begin{enumerate} \item[\bf a)] For all $n \in \mathbb N$, we have that [\sum_{k=1}^n k^3 = \frac{n^2(n + 1)^2}{4}.] \item[\bf b)] Show that for $x, y \in \bZ$ and $n \in \bN$ (with $x \neq y$), we have that [x - y \text{ divides } x^n - y^n] \item[\bf c)] Show that for all $n \in \bZ$, $5$ is a divisor of $2^{3n} - 3^n$ \item[\bf d)] For all $n \in \bN$ with $n \geq 5$, we have that [n! > 2^n > n^2] \end{enumerate}\end{problem} \emph{As a hint to {\bf c)}: $x^{n + 1} - y^{n + 1} = x^{n + 1} - xy^n + xy^n - y^{n + 1}$.}
void yew
```
k^3 + 3k^2 + 8k
k^3 + 3k^2 + 9k - k
k^3 -k + 3k^2 + 9k
(k-1)(k)(k+1) + 3k(k ...
add a couple more lines to show proof for how both sides are divisible by 6
final swift
Okay I might do that at home since I dont want to miss the bus
void yew
welp I'm happy with my 4 line solution now
final swift
What is z again?
shut obsidian
void yew
i would'nt call my method induction ig, since I did some simplification and found the result
shut obsidian
same pls, i wanna practice more
There
shut obsidian
Yep
void yew
the problem is resolved @final swift
final swift
.close
torpid tapir
topaz sinewBOT
torpid tapir
how do i solve this?
acoustic tangle
Have you tried anything so far?
torpid tapir
yeah
wooden moon
what did you try?
acoustic tangle
Could you show your work then?
acoustic tangle
So far so good
Not having a fraction on the right hand side would be nice as well, consider multiplying both sides by the denominator on the right hand side
torpid tapir
yeah thats what im trying to do
devout veldt
Are you trying to solve for x or just factor and simplify?
torpid tapir
solve for x
the method im trying is really long
ik a way where you make everything have the same denominator but i dont fully know how to do that
devout veldt
ik a way where you make everything have the same denominator but i dont fully kn...
I mean you could subtract 3x/x-1 over to the left then subtract (x^2+4)/1 - the (3x)/(x-1)
torpid tapir
Does this work?
acoustic tangle
Yeah
torpid tapir
its the most understandable imo
devout veldt
whatever works for you is how you should do it
devout veldt
its the most understandable imo
do you need the final equation to be x = ? or is having x^2 fine?
torpid tapir
I’m done I js have to eliminate the options for x
acoustic tangle
Factoring one side of an equation when solving for a single real unknown is mostly useful only if the other side is zero
devout veldt
I’m done I js have to eliminate the options for x
your x values will just be 4 and 4/5
topaz sinewBOT
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harsh falcon
How would you determine convergence or divergence?
topaz sinewBOT
How would you determine convergence or divergence?
harsh falcon
I seem to be left with nth(sqrt(n^n-n^2)
glacial adder
$\lim_{n\to\infty}\left(\frac{n^n}{n^{n^2}}\right)^\frac{1}{n}$
thorny flameBOT
y0shi
harsh falcon
is it also possible with harmonic series for 1/2 summation 1/n? making r=-1/5? obviously more tedious but im just checking
harsh falcon
**y0shi**
aslo this means that its technically inconclusive but since the 1/2n dominates -1^n/5^n that its divergent?
glacial adder
is it also possible with harmonic series for 1/2 summation 1/n? making r=-1/5? o...
not quite sure what you’re trying to say here
the series is nowhere close to a harmonic series
harsh falcon
what does the limit evaluate to?
1?
thorny flameBOT
y0shi
harsh falcon
I didnt remove the proper exponent from the numerator, but I see now how it is removed
glacial adder
alright
harsh falcon
.close
timber tulip
What am I doing wrong and how can I fix it? I’m trying to find the second deravtibes of y^4+5x=11
topaz sinewBOT
What am I doing wrong and how can I fix it? I’m trying to find the second deravt...
timber tulip
I found the first deravative of it which was -5x/(4(y)^3)
After that, I took the deravative or the second
So it would be -5 (4(y)^3) - d/dx (4(y)^3) (-5x) all over 16y^6
Oh
The first derevative is not that, it’s actually -5/4(y^3)
I have trouble finding the second deravtibe
topaz sinewBOT
@timber tulip Has your question been resolved?
bitter wigeon
hi
bitter wigeon
how was it possible to add -3
hollow plover
Because a² - ab + b² = a²+2ab + b² - 3ab = (a+b)² - 3ab
a = sinx and b = cosx in your case
@bitter wigeon Ok ?
topaz sinewBOT
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mystic siren
how do i do this algebric proof?
topaz sinewBOT
how do i do this algebric proof?
mystic siren
for a,b>=0, and in R
prisma mesa
I'd just assume a >= b without loss of generality, and prove it holds
(If you know that a >= b, then you can "evaluate" both the min and max)
mystic siren
(If you know that a >= b, then you can "evaluate" both the min and max)
so i can say
let a>=b, therefore:
a>=b -> -a<=-b
and because max(a,b) returns the biggest number of the two parameters, and min(a,b) the minimum number of the two. therefore we can conclude in this case max(a,b=a, and min(-a,-b)=-a, according to the functions max and min's definitions.
for the case of b>a the oppsite will be true.
q.e.d.
do you think this is good enough?
@prisma mesa
prisma mesa
I'd also include something like "hence min(-a, -b) = -max(a,b)" after evaluating max and min
mystic siren
I'd also include something like "hence min(-a, -b) = -max(a,b)" after evaluating...
yes i forgot to refer to the original statement, ty
prisma mesa
and instead of saying "for the case of b > a the opposite will be true", I'd just use the phrase
"Without loss of generality, assume a >= b" at the start
because it doesnt really matter whether we call the first entry a or b
So like replace "let a>=b, therefore" with "Without loss of generality, assume a >= b"
mystic siren
So like replace "let a>=b, therefore" with "Without loss of generality, assume a...
it does feel that we are losing generality though?
prisma mesa
Or instead of saying "the opposite will be true", use something like "similar proof shows that min(-a, -b) = -max(a,b)", because "opposite" is unclear i thnik
prisma mesa
it does feel that we are losing generality though?
We dont.
If a <= b, then we can just swap a and b in the min and max and it will become
min(-b, -a) = -max(a, b) and b >= a, which is exactly the same scenario as we're proving
mystic siren
Or instead of saying "the opposite will be true", use something like "similar pr...
i think i understand
prisma mesa
but ig you can avoid that phrase by considering the a < b case
mystic siren
We dont.
If a <= b, then we can just swap a and b in the min and max and it wil...
oh i understand
mystic siren
but ig you can avoid that phrase by considering the a < b case
i think i know how to write it now
ty mate
!close
oh well
topaz sinewBOT
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Available help channel!
deft lark
so, my friends and i are doing a math competition and the question is x^100=1 but in complex numbers and the answer is supposed to be in the format of X.xxxxxxxxxx +- X.xxxxxxxxxx where capital x is a zero he says that there are multiple answers but only one is the one he's looking for... i have absolutely no clue how to do this as i'm still in high school. if anyone wanted to help me understand and beat the challenge that would be appreciated , i understand if not, i'll leave this open for a while and then close it in a couple of hours, thanks
topaz sinewBOT
so, my friends and i are doing a math competition and the question is x^100=1 bu...
dense rain
A lot of math competitions, depending on the level of prestige expect that you train for it and have seen things outside of your standard curriculum.
This is a standard roots of unity problem, but I'm not sure what is so special about these numbers that you could easily see 1 obvious answer out of all of them without doing the problem myself
cinder sequoia
okay, firstly you should understand that there are 100 complex solutions to x^100 = 1, but it sounds like the form you gave is for a single particular real root?
the only two purely real roots are -1 and 1
dense rain
They said it has to have zero interger parts for the real and imaginary component
cinder sequoia
oh okay so it's one of the 98 complex roots
dense rain
I'm curious which one is the interesting one among them, but I'm not going to do this lol
cinder sequoia
that's weird though, all of those 98 roots fit the form cos(theta) +/- i * sin(theta) so almost all should theoretically have 0 integer parts
are the numbers of xxxxx in front of the formatted solution of significance?
like does that mean the solutions all have the same digit or are those just placeholders
cool visual btw lmao
topaz sinewBOT
@deft lark Has your question been resolved?
deft lark
cool visual btw lmao
so, it's one of the points on that plot? with the y value being i or imaginary number
deft lark
like does that mean the solutions all have the same digit or are those just plac...
placeholders, sorry i should have known that i would have to clarify haha
topaz sinewBOT
@deft lark Has your question been resolved?
deft lark
<@&286206848099549185>
topaz sinewBOT
@deft lark Has your question been resolved?
keen venture
Doesn't seem like there's an actual answer to this question
deft lark
there is
are*
there are just a lot of answers
as seen in the diagram above
but i need all 98 answers and i don't know that graph so i can't get it
i'll try some stuff in desmos but idk that's all i can figure out
honest trellis
sketch 2/x^2+1
topaz sinewBOT
sketch 2/x^2+1
honest trellis
could someone please help me with this question
i know how to sketch 1/x^2 +1 but i'm not really sure if theres a 2 as the numerator
deft granite
i know how to sketch 1/x^2 +1 but i'm not really sure if theres a 2 as the numer...
Do you know about asymptotes?
honest trellis
yes
i typed the graph up on desmos
and it looked so
idk like unconventional
why does its shape look like a mountain
deft granite
Can you show a screenshot?
honest trellis
deft granite
Can you show the original task? Is it 2 / (x^2 + 1) or (2 / x^2) + 1?
honest trellis
Can you show the original task? Is it `2 / (x^2 + 1)` or `(2 / x^2) + 1`?
sorry i cant send it in , the question is on a book and i dont have a phone
☹️
deft granite
was the function they wanted me to sketch
So you know how to sketch 1 / (x^2 + 1)?
honest trellis
So you know how to sketch `1 / (x^2 + 1)`?
nope , sorry
i take that back
i thought it would look the the graph with two of those l looking shapes at an asymptote
and like a concave down parabola
i did not expect to see this weird looking shape
deft granite
First, is 2 / (x^2 + 1) continuous? Does it have any "breaks" like 1 / x^2 does?
deft granite
i have no idea what continuous means
Is there any x where there's no value for 2 / (x^2 + 1)?
honest trellis
Is there any `x` where there's no value for `2 / (x^2 + 1)`?
i think probably no
deft granite
right, the denominator can never become 0
Is this function symmetric across the y axis?
honest trellis
yes
waitt so if it has no asmyptotes its symmetric about y axis?!
thank you so much for helping btw
deft granite
waitt so if it has no asmyptotes its symmetric about y axis?!
🤔 wdym
honest trellis
Why do you think so?
i have no idea i think i looked at the desmos graph
deft granite
Well, the point of the task is to get a rough idea of the graph without looking at the precise graph
Do you know how to find out whether the graph is symmetrical? What does the graph being symmetrical mean?
deft granite
An even function is symmetric across the y axis, yes.
How do you figure out if a function is even?
deft granite
well, there are usually several things to consider when making a sketch, like
- whether a function is even/odd
- whether there's any other symmetry (e.g.
1/(x-10)is neither even or odd, but still has symmetry aroundx = 10) - does the function have any asymptotes?
- are there any interesting points in the function? e.g. in
1/xit's 0, insqrt(2 - x)it's 2 where the function ends
If you're really stuck, you can compute the function at several points and try connecting them
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Available help channel!
fallow pivot
I’m asked to evaluate but I forgot how to
topaz sinewBOT
I’m asked to evaluate but I forgot how to
cinder sequoia
yes
topaz sinewBOT
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Available help channel!
worn gorge
topaz sinewBOT
worn gorge
im not rly sure which expression i should try
gleaming reef
oh
factor out the n and think about this combinatorially
in particular, does this count the subsets of [n-1]
or just apply binomal thm
worn gorge
idk i cant find it
no cos
i think ur meant to use a reasonable substitution
like
(1+x)^n
or smth
but i cant figure out
which one to use
worn gorge
2 to the power of n-1
gleaming reef
2 to the power of n-1
binomial expansion
worn gorge
gleaming reef
good job chef
worn gorge
and then
gleaming reef
now multiply it by n
worn gorge
u see
gleaming reef
what's (n-1 choose n-1)
worn gorge
its 1
teal gust
it only goes up to n-2
so subtract n-1Cn-1
teal gust
from the total sum
gleaming reef
yeah just subtract 1
gleaming reef
@worn gorge
worn gorge
okok
gleaming reef
ty for using this service
gleaming reef
lemme c if it
ofc
grand solar
ah nvm
autumn plaza
topaz sinewBOT
autumn plaza
what does question 21 mean
cursive patrol
do (AB)C and A(BC) and note that the results are equal
cursive patrol
yes
autumn plaza
Okok
lucid junco
Bum chicken
topaz sinewBOT
@autumn plaza Has your question been resolved?
autumn plaza
.close
topaz sinewBOT
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mortal mirage
Can someone pls tell me if I’m correct
Oh yeah ur right
I forgot x was there
but what about the other ones?
Oh so no brackets
and in the middle would be - right?
uh
never really heard of it..
Oh
I also solved this
7
-3
ohhh
-7
I forgot we need to take higher numbers sign
oh
let’s go
Can u tell me how to solve this?
don’t we have to multiply the question by 2?
but what’s the correct way of doing it?
So both can be correct
Taht also includes multiplying c by 2 right?
So after we multiply them what to do?
lemme send a pic
thorny flameBOT
faiyrose
mortal mirage
**faiyrose**
Oh so now the 2 cancel each other?
Right?
okay one sec
I got this
do we have to move d3 with 2c
Then it would be d^3=2c+10
it would be
wait
-d^3?
wait let me try to do it
I’m kinda lost
don’t we have it -10 from the whole thing
Now I got 2c-10=d^3
there’s a concept I do
know
Where like
u take something that cancels with something
I don’t rlly remember it
Yeah
thorny flameBOT
faiyrose
thorny flameBOT
faiyrose
mortal mirage
ohh
what about the the 2nd one
Cube root cancels with cube right?
So we cube the whole equation?
Would be v^3=cube root p^3+r^3
Now what
thorny flameBOT
faiyrose
mortal mirage
Yup
oh
but what about v?
v^3 rigjt
Oh so
that cube cancels
with it
now wouldn’t it be = v^3=p+r
Now we should subtract p from the whole thing right
Now I got
V^3-p=r
Now what
But we need to make r the subject
Ohh
3rd one
I’m lost with this
If I had to guess I would multiply whole equation by g
Ohh
Expand
2mh=g1-gh?
oh
We take common
h
So it would be like h(2m-g1-g)?
I think that’s wrong
Oh wait
I did a mistake
when expanding
g multiply by 1
Would still be
g
1
so it would be 2mh=g-gh
there
oh so 2mh+gh=g
now we take common
so h(2m+g)=g
now we just divide both sides by h(2m+g)
so h =g/2m+g
what abt this
is that l or e
lets say it was e
how do we solve?
i dont think questions lie this wil come
cuz my teacher ssaid such questions wont come
is this corrwect
ph
oh
ahem
thought
it was 2
its 15
mb
so ans is 7
how to do this?
thats it
one sec
adn then we simplify?
yep thats easy
is this corect
how to solve thid
mortal mirage
should the 1 be moved before equating the denominators
carmine jungle
No
carmine jungle
can someoe solve step by step
Multiply the numerator and denominator of 3(3 - x) by 5
So you have 15(3-x)/5
topaz sinewBOT
@mortal mirage Has your question been resolved?
topaz sinewBOT
topaz sinewBOT
Available help channel!
mortal mirage
how to solve
topaz sinewBOT
how to solve
mortal mirage
acoustic pecan
what do you think
mortal mirage
what do you think
i dont know.
acoustic pecan
what do you think "y varies as the cube of (x+2)" means
topaz sinewBOT
@mortal mirage Has your question been resolved?
paper robin
a from where??
we can just use it a like another value we r tryna find to get y
so taking y=a(x+2)^3 for example
we can substitute our values into the equation
32=a(0+2)^3
32=(a)^3
paper robin
all g
mortal mirage
paper robin
immediatley from this question, we can name a bunch of angles already
mortal mirage
yep
mortal mirage
im guessing 2x
mortal mirage
yep
paper robin
and now
now that we have found all that out
2x+5x+x=180
why 180?
because its the sum of all angles = 180
mortal mirage
oh
paper robin
go ahead
mortal mirage
mortal mirage
yeah
mortal mirage
oh yeah
paper robin
i hope im not mistaken
mortal mirage
what about x