imqwerty
  • imqwerty
emergency!!!
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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jamiebookeater
  • jamiebookeater
I got my questions answered at brainly.com in under 10 minutes. Go to brainly.com now for free help!
imqwerty
  • imqwerty
ganeshie8
  • ganeshie8
which q
ParthKohli
  • ParthKohli
Last one x = y = z = 4

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imqwerty
  • imqwerty
any question that u like...these r the question which i wsn't able to solve
Empty
  • Empty
First one probably some fermat's little theorem nonsense, the second one looks like part of the Riemann zeta function so you could probably just multiply two geometric series or something together for it, and 3rd one looks like it's probably a telescoping series, just my first guesses
imqwerty
  • imqwerty
@ParthKohli how did u get x=y=z=4
ParthKohli
  • ParthKohli
It's a symmetric situation. Apply all inequalities you please, but you'll finally get to this.
Empty
  • Empty
Simplest way to think of it (not a proof) is that \[n^2 > (n-a)(n+a)\] when a is not equal to zero, so that's the max when they're all the same.
ganeshie8
  • ganeshie8
first q doesnt look easy
Empty
  • Empty
I think we can split that 3rd one up using partial fractions, then it becomes a telescoping series that way.
Empty
  • Empty
Good questions by the way!
imqwerty
  • imqwerty
thanks @Empty @ParthKohli i can't get to the result x=y=z. can u tell how to start ..
ParthKohli
  • ParthKohli
I've only given a qualitative argument here so far - the whole situation you have here is symmetric in nature. x can be interchanged with y, y with z or z with x, and the situation won't change. So if the triplet \((a, b , c)\) gives you a maximum value then so should \((b, a, c)\) or \((b, c, a)\) or \((c, a, b)\), which means that \(x = y = z \) when the given expression is maximum. Let me think of an inequality that can be applied here. You'll see that we'll find the maximum when the equality of two expressions is considered, which is almost always when all variables are equal.
Empty
  • Empty
Ok so for the second one I'm getting: $$\frac{1}{1-9}\frac{1}{1-4}\frac{5}{6}$$ Does that match your answer?
imqwerty
  • imqwerty
wait m doing the 1st one
ganeshie8
  • ganeshie8
for first one you may want to try factoring it http://www.wolframalpha.com/input/?i=factor+5%5E5%5E%28n%2B1%29%2B5%5E5%5En%2B1
Empty
  • Empty
Yeah I think it is more clear how to solve the first one if you do that, here to make factoring easier substitute: $$5^{5^n}=a$$ then you end up with the expression: $$a^5+a+1$$ which somehow wolfram is able to factor out into: $$(a^2+a+1)(a^3-a^2+1)$$
imqwerty
  • imqwerty
@ParthKohli if we are given a question like - x+y=4 and x belongs to positive integers. and we are supposed to find out maximum of xy + x/y +y/x then by symmetric approach x=y=2 but in this question x=3 and y=1 gives the maximum so the symmetric approach is violated.
imqwerty
  • imqwerty
@Empty we can't write 5^5^(n+1) as a^5 if a=5^5^n
ParthKohli
  • ParthKohli
OK, I didn't say that x = y gives you the maximum. I said it gives you the maximum if it exists. Here, x = y is most probably giving you the minimum.
imqwerty
  • imqwerty
yes so how do we know that in which case x=y gives the max
Empty
  • Empty
$$5^{5^{n+1}}=5^{5*5^n} = (5^{5^n})^5$$ @imqwerty
ParthKohli
  • ParthKohli
You have to check if x = y is giving you the maximum/minimum by entering another pair of values. If that pair gives you a higher value, then x = y gives you the minimum.
ganeshie8
  • ganeshie8
i feel #1 is still a stupid q, you're not gona learn anything frm it
imqwerty
  • imqwerty
wait sry yes we can put a^5...
imqwerty
  • imqwerty
yes it is @ganeshie8 :D
ParthKohli
  • ParthKohli
Y'know, qwerty, it's just like that with single variable functions. You can differentiate a function and equate it to zero, but you still have to check if that gives you the maxima or minima by other methods.
Empty
  • Empty
Ok so I also found a good way of solving the second one: We only want odd total powers, so the way we can do that is only combine even powers with odd and odd with evens. So in order to do that, we multiply the geometric series' together: $$\left(\sum_{i=0}^{\infty} \frac{1}{2^{2i}} \right) \left(\sum_{j=0}^{\infty} \frac{1}{3^{2j+1}} \right) +\left(\sum_{i=0}^{\infty} \frac{1}{3^{2i}} \right) \left(\sum_{j=0}^{\infty} \frac{1}{2^{2j+1}} \right) $$ factor out a 1/3 and 1/2 so you get only geometric series in terms of 1/4 and 1/9, and you're done!
ParthKohli
  • ParthKohli
Isn't that the standard way to do it though? How did you originally do it?
Empty
  • Empty
I factored it
imqwerty
  • imqwerty
the 1st one is solved thanks guys :)
ganeshie8
  • ganeshie8
\[\begin{align} x^5 + x + 1 & = (x^5-x^2)+(x^2+x+1)\\ \\ & = x^2(x^3 - 1) + (x^2 + x + 1) \\ \\ & = x^2(x-1)\color{blue}{\bf (x^2 + x + 1)} + \color{blue}{\bf (x^2 + x + 1)}\end{align}\] http://math.stackexchange.com/questions/477295/factor-x5-x-1
ParthKohli
  • ParthKohli
To be fair, that solution was most likely obtained by working backwards. :P
ganeshie8
  • ganeshie8
that is okay for a proof
ParthKohli
  • ParthKohli
Yes of course, but you can't really think of that on your own.
ganeshie8
  • ganeshie8
that all goes as "scratch work" :P
Empty
  • Empty
ahhh thanks @ganeshie8 that's a clever trick I think I can even remember that as adding fancy form of zero and then abusing the geometric series yet again lol
ParthKohli
  • ParthKohli
I'm ashamed that I have no other ideas other than telescoping for #3.
Empty
  • Empty
It wants to telescope, why force it to be anything it's probably not? :P
Empty
  • Empty
When you separate it out, the coefficients are both 1, it's truly a beautiful result
ganeshie8
  • ganeshie8
I'm either dumb or too much drunk to notice it but why do you think telescoping is a bad idea ?
Empty
  • Empty
Yeah you're not drunk enough ganeshie, cause this baby wants to see the moon
ParthKohli
  • ParthKohli
I didn't say telescoping is a bad idea. I just said that this series doesn't seem to be telescoping nicely and I have no other ideas.
imqwerty
  • imqwerty
m really confused nd annoyed nd kinda drunk ..cheers @ganeshie8 @Empty ..can u please tell me what u did with the 2nd question
imqwerty
  • imqwerty
my mind is jst nt wrkin normal today :(
ParthKohli
  • ParthKohli
OK, it's actually telescoping really nicely.
Empty
  • Empty
Here I made it smaller, part of the total sum, just write out all these terms (it's not tooo many, and it's good for you) \[\left(\sum_{i=0}^{3} \frac{1}{2^{2i}} \right) \left(\sum_{j=0}^{3} \frac{1}{3^{2j+1}} \right) +\left(\sum_{i=0}^{3} \frac{1}{3^{2i}} \right) \left(\sum_{j=0}^{3} \frac{1}{2^{2j+1}} \right) \]
imqwerty
  • imqwerty
no i mean hw did u split that
Empty
  • Empty
No, I _created_ this
Empty
  • Empty
so by multiplying it out you can kinda see how I came up with this too... I'll draw a picture of how I think about it
ParthKohli
  • ParthKohli
\[\frac{6^k}{(3^k - 2^k)(3^{k+1 } - 2^{k+1})}\]\[= \frac{3^{k+1}}{2^{k+1} - 3^{k+1}} - \frac{3^k}{3^k - 2^k}\]
ParthKohli
  • ParthKohli
There's a typo.
Empty
  • Empty
|dw:1438453754914:dw| So this is the first half of the terms! @imqwerty
ParthKohli
  • ParthKohli
Supposed to be\[= \frac{3^{k+1}}{2^{k+1} - 3^{k+1}} - \frac{3^k}{2^k - 3^k}\]
Empty
  • Empty
Specifically I knew to write that because the only way for i+j=odd is when one is even and the other is odd since odd+odd=even and even+even=even. So combine all the 2s with even powers with all the 3s with odd powers, and add it to all the 2s with odd powers with all the 3s with even powers.
Empty
  • Empty
so that's where the 2i and 2i+1 is coming from, keeping the powers even and odd: \[\left(\sum_{i=0}^{\infty} \frac{1}{2^{2i}} \right) \left(\sum_{j=0}^{\infty} \frac{1}{3^{2j+1}} \right) +\left(\sum_{i=0}^{\infty} \frac{1}{3^{2i}} \right) \left(\sum_{j=0}^{\infty} \frac{1}{2^{2j+1}} \right) \]
imqwerty
  • imqwerty
thank u so much @Empty :)
ParthKohli
  • ParthKohli
OK, so we're sorta done with all of the questions.
ganeshie8
  • ganeshie8
whats the ansur for q2
Empty
  • Empty
Are you asking me or asking imqwerty?
imqwerty
  • imqwerty
m calculating the answer...
ParthKohli
  • ParthKohli
@Empty Did you check out the solution to the problem I gave you yesterday?
Empty
  • Empty
Oh no I didn't, also I ended up getting 2 to that, I'll go check real fast
ParthKohli
  • ParthKohli
2 is correct.
imqwerty
  • imqwerty
:) a big THANKS TO ALL OF YOU @Empty @ParthKohli @ganeshie8 :)
ganeshie8
  • ganeshie8
#2 im getting 5/144
ParthKohli
  • ParthKohli
Ganeshie and Qwerty haven't done that problem so I'll ask them to do it too.\[\sum_{n=1}^{\infty}\frac{F_n}{2^n}\]where \(F_n\) is the \(n\)-th Fibonacci number. \(F_1 = F_2 = 1 \).
imqwerty
  • imqwerty
my answer the #2 is 121/144
ganeshie8
  • ganeshie8
I am familiar with that problem \[\sum_{n=1}^{\infty}F_nx^n= \frac{x}{1-x-x^2}\] plugin \(x=1/2\)
Empty
  • Empty
Here's the solution: \[\left(\sum_{i=0}^{\infty} \frac{1}{2^{2i}} \right) \left(\sum_{j=0}^{\infty} \frac{1}{3^{2j+1}} \right) +\left(\sum_{i=0}^{\infty} \frac{1}{3^{2i}} \right) \left(\sum_{j=0}^{\infty} \frac{1}{2^{2j+1}} \right) \] \[\left(\sum_{i=0}^{\infty} \frac{1}{2^{2i}} \right) \frac{1}{3}\left(\sum_{j=0}^{\infty} \frac{1}{3^{2j}} \right) +\left(\sum_{i=0}^{\infty} \frac{1}{3^{2i}} \right) \frac{1}{2} \left(\sum_{j=0}^{\infty} \frac{1}{2^{2j}} \right) \] \[\left( \frac{1}{3} + \frac{1}{2} \right) \left(\sum_{i=0}^{\infty} \frac{1}{3^{2i}} \right) \left(\sum_{j=0}^{\infty} \frac{1}{2^{2j}} \right) \] \[\left( \frac{1}{3} + \frac{1}{2} \right) \left( \frac{1}{1-3^2} \right) \left(\frac{1}{1-2^2} \right) \] \[\frac{5}{144}\]
imqwerty
  • imqwerty
damn my mind is nt ok today ....instead of starting with i=0 i started with i=1. :( feeling bad at the silly mistakes m doin :(
Empty
  • Empty
Haha nah it's fine that's why I wrote it all out for you. And hey, if you want you can always come back tomorrow and look, no reason to feel bad XD
Empty
  • Empty
I'm pretty sure "feeling bad" is not supposed to be a step in solving problems haha, even though sometimes it does tend to be when those problems are proofs in real analysis for me lolol
ganeshie8
  • ganeshie8
sometimes i hate problems from indian competitive exams/tests mostly because they are meaning less, how do they expect a 10/12th grader to know about generating functions and stuff ?
ganeshie8
  • ganeshie8
they just want the exams to be tough, they dont bother to look at the syllabus of 12th grader
imqwerty
  • imqwerty
the reason y m feeling bad and annoyed - today the results of the last test held at of coaching institute came- the 1st ranker got 360/360 nd i ws like O_o nd i ws 12th with score 329. i could've done way better if i wuld have rechecked my paper after completing it instead of looking at other ppl faces :(
imqwerty
  • imqwerty
at my home m nicknamed - 'silly mistake'
ganeshie8
  • ganeshie8
trust me, you and PK are way better than me when i was in my 12th
Empty
  • Empty
Yeah I don't understand the obsession with rote memorizing all these useless one-time-use tricks for problems mankind invented calculators and wolfram alpha for. Time to change the curriculum to a modern age I think.
imqwerty
  • imqwerty
@Empty so true^ .
Empty
  • Empty
I always see these questions like "find the number of roots of this polynomial" or "show this ridiculous trig identity is true without any geometric picture that it came from" ...why?
Empty
  • Empty
It's like teachers want people to believe math is useless AND ugly.
ParthKohli
  • ParthKohli
Whut, qwerty? The top-ranker got 360?! Surely the questions must have been really simple. The topper at my coaching (me) got 307/360 which was an exception to the rule that the previous years' toppers got no more than 250-260.
ParthKohli
  • ParthKohli
humblebrag
Empty
  • Empty
lol duuuude gotta go for that \(2 \pi\) score man!
imqwerty
  • imqwerty
the paper wasn't enough easy to get 360/360. the guys are too crazy...they are like 24 x 7 studying.
Empty
  • Empty
Their parents must be awful people
Empty
  • Empty
Ok ok maybe I'm being pessimistic here haha
imqwerty
  • imqwerty
The guy who got 1st rank is a vry good frnd of mine. He says his parents think that he can do whatever he wants but he must remain healthy eating proper food.
imqwerty
  • imqwerty
his parents don't interfere his personal life
Empty
  • Empty
Woah he must be a genius :O
imqwerty
  • imqwerty
this guy got 1st rank in 2/4 test held till date
ParthKohli
  • ParthKohli
Kaunsa institute hai ye?
ganeshie8
  • ganeshie8
see you cant study more than 14 hours each day, you need at least 6 hours of sleep and 4 hours for maintenance
imqwerty
  • imqwerty
allen
ParthKohli
  • ParthKohli
Kota wala?
imqwerty
  • imqwerty
han
ParthKohli
  • ParthKohli
Tumhe internet mil jaata hai?
imqwerty
  • imqwerty
mai to kota ka hi hu
ParthKohli
  • ParthKohli
Oh, badiya yaar.
imqwerty
  • imqwerty
:) badiya bhi hai aur thodi prblems bhi hai....roz ghar par koi na koi guest aa jate hai...roz tv etc etc ka shore distrb karta hai...nd many more prblems
ParthKohli
  • ParthKohli
Guest kyun aa jaate hain?
imqwerty
  • imqwerty
pata nai... btw mere relatives paas me hi rahte hai to wo dadi dada se milne aa jate hai.
imqwerty
  • imqwerty
btw @ganeshie8 did u got the answer to the question - dIzI/dz z=complex number
ganeshie8
  • ganeshie8
common sense : common lannguage guys, when others are there in the thread
imqwerty
  • imqwerty
ok :) sry
ganeshie8
  • ganeshie8
thats okay, just saying..
ParthKohli
  • ParthKohli
あなたは何について話していますか?理解できません。
ganeshie8
  • ganeshie8
dIzI/dz derivative doesnt exist
ParthKohli
  • ParthKohli
What kind of derivative is this?
ganeshie8
  • ganeshie8
z is a complex number f(z) = |z| is a real valued function that takes complex number as input
ParthKohli
  • ParthKohli
Yeah, I can see that. But it's a complex number. How do you compute derivatives of complex-valued functions? Is it that instead of taking a small interval, you take a small circular area?
ganeshie8
  • ganeshie8
Exactly, the "interval on real number line" becomes a "ball in argand plane"
ParthKohli
  • ParthKohli
@imqwerty knows all this?! o.O
imqwerty
  • imqwerty
not too much..
ParthKohli
  • ParthKohli
Well, there goes my self-esteem.
ganeshie8
  • ganeshie8
if you're in 12th grade and if you say you know all this, then you're just lying
ParthKohli
  • ParthKohli
He's in 11th grade!
ParthKohli
  • ParthKohli
So am I.
EclipsedStar
  • EclipsedStar
Me too...lol
imqwerty
  • imqwerty
i asked my teacher to explain this but he refused :( he said - its of no use
ParthKohli
  • ParthKohli
That's stupid. "It's not there in the JEE examination" should not be restated as "it's of no use".
imqwerty
  • imqwerty
^true that
ParthKohli
  • ParthKohli
I think complex derivatives are a hell lot more useful that the four questions you gave us.
imqwerty
  • imqwerty
yes!! the teaching methodology nd the mindset of ppl must change.
ganeshie8
  • ganeshie8
hmm your teacher knows more than you, he might be thinking that teaching you about complex derivatives is useless because you ppl are not ready yet
ParthKohli
  • ParthKohli
What if he straight-up doesn't know what they are? That's a possibility too. I don't know much about Allen teachers but I don't expect mine to give a satisfactory explanation.
imqwerty
  • imqwerty
es this can be the reason @ganeshie8 nd we nt taught complex numbers till nw ..but still he said that its far away frm JEE syllabus. but i'll ask him when the chapter gets over. I asked Anna sir [HOD math allen] nd m sure he knws it
ParthKohli
  • ParthKohli
Haha, of course it's beyond the reaches of JEE syllabi. It's not just the complex numbers we're taught in 11th and 12th.
ParthKohli
  • ParthKohli
Wow, I just searched him up and he looks pretty young, like a passout.
imqwerty
  • imqwerty
m enrolled in vibrant academy too i'll go and ask Vikas gupta sir (HOD math vibrant).i think he will gimme the explanation.
ParthKohli
  • ParthKohli
I think it's better to not ask for it...
ganeshie8
  • ganeshie8
If you get a satisficatory explanation, please do share wid us
imqwerty
  • imqwerty
haahaha once i got kicked out (frnkly) fr asking such doubts.. nd yes i'll definately share it.
imqwerty
  • imqwerty
pk have u joined any institute?
ParthKohli
  • ParthKohli
Yeah, Vidyamandir.
imqwerty
  • imqwerty
oh nice :)
anonymous
  • anonymous
consider \(x^2+x+1\) divides into both \(x^5+x+1,x^{25}+x^5+1\). this is sufficient to prove \(5^{5^{n+1}}+5^{5^n}+1\) is not prime for positive integer \(n\)
anonymous
  • anonymous
for the next problem, you just break it into an iterated sum, though if you don't set a lower bound on \(i,j\) individually you will run into problems
imqwerty
  • imqwerty
thanks @oldrin.bataku heres the new question that m solvin -
1 Attachment
ParthKohli
  • ParthKohli
Eh, this problem isn't that hard.
ParthKohli
  • ParthKohli
It may be calculative, but it's not hard.
ganeshie8
  • ganeshie8
looks that fraction evaluates to \(\left(\dfrac{n+1}{n}\right)^2\)
imqwerty
  • imqwerty
Yes it has lil bit calculation prt
Empty
  • Empty
You can do the gauss fliperoo method there I think.
anonymous
  • anonymous
$$\sum_{k=1}^n (2k)^2=4\sum_{k=1}^n k^2=4\cdot\frac{n(n+1)(2n+1)}6\\\sum_{k=0}^n(2k+1)^2=\sum_{k=0}^n (4k^2+4k+1)=4\sum_{k=1}^n k^2+4\sum_{k=1}^n k+\sum_{k=0}^n 1$$ so the second sum gives us: $$4\cdot\frac{n(n+1)(2n+1)}6+4\cdot\frac{n(n+1)}2+(n+1)$$
anonymous
  • anonymous
oops, the second should be the sum up to \(n-1\), actually
Empty
  • Empty
@imqwerty You might like Mathematical Methods in the Physical Sciences by Mary Boas. I think you might like it, it introduces complex numbers and complex analysis in separate chapters, check it out: https://faculty.psau.edu.sa/filedownload/doc-4-pdf-0a187866618ca3049030ec5014860ae8-original.pdf It's very accessible and if you have any questions feel free to ask, that book covers so much more though than just that.
imqwerty
  • imqwerty
ok i am getting 151 as the answer
imqwerty
  • imqwerty
thank u so much @Empty ^^
imqwerty
  • imqwerty
wait no m getting 150 as the answer
anonymous
  • anonymous
so $$\frac23\cdot n(n-1)(2n-1)+2 n(n-1)+n$$ which reduces the rational expression as $$\frac{2n+2}{2n-1}>1.01\\2n+2>1.01(2n-1)\\2+1.01>2(1.01-1)n\\3.01>0.02n\\n<50\cdot 3.01=150.5$$ so \(n\le 150\)
imqwerty
  • imqwerty
correct @oldrin.bataku :)
anonymous
  • anonymous
for the last one \(xyz+xy+yz+zx\) links us to $$(x+1)(y+1)(z+1)=(x+1)(yz+y+z+1)\\\quad =xyz+xy+zx+yz+x+y+z+1$$ so we ahve that $$(x+1)(y+1)(z+1)-(x+y+z)-1=xyz+xy+zx+yz$$ now, we know \(x+y+z=12\) so: $$(x+1)(y+1)(z+1)-13=xyz+xy+zx+yz$$ now, let's use the AM-GM on \(x+1,y+1,z+1\): $$\frac{(x+1)+(y+1)+(z+1)}3\ge \sqrt[3]{(x+1)(y+1)(z+1)}$$ so it follows: $$(x+1)(y+1)(z+1)\le \left[1+\frac{x+y+z}3\right]^3$$ which gives us: $$(x+1)(y+1)(z+1)\le \left[1+4\right]^3=125$$ so we have that $$(x+1)(y+1)(z+1)-13\le125-13=112$$
imqwerty
  • imqwerty
thank u so much!! @oldrin.bataku now all the questions r solved ^~^
imqwerty
  • imqwerty
@Empty in the second question in the last step u took the common ratios of the 2 infinite GPs wrong
anonymous
  • anonymous
complex differentiation is not beyond what a high schooler can know; I used the residue theorem in calculus II which was in 11th grade
anonymous
  • anonymous
anyways, \(|z|=z\bar z\) is not differentiable on \(\mathbb{C}\) because it depends on \(\bar z\)
anonymous
  • anonymous
$$\sum_{k=0}^N\left(\frac{3^{k+1}}{2^{k+1} - 3^{k+1}} - \frac{3^k}{2^k - 3^k}\right)=\frac{3^{N+1}}{2^{N+1}-3^{N+1}}-\frac{3^0}{2^0-3^0}$$ the first term behaves like: $$\frac{1}{(2/3)^{N+1}-1}\to \frac1{-1}=-1$$as \(N\to\infty\)
anonymous
  • anonymous
also it should be \(k=1\) giving: $$-\frac{3^1}{2^1-3^1}=-\frac{3}{2-3}=-\frac{3}{-1}=3$$ so \(-1+3=2\)
anonymous
  • anonymous
@imqwerty
imqwerty
  • imqwerty
thanks @oldrin.bataku ^~^ :)
imqwerty
  • imqwerty
@dan815

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