#Binomial theorem

93 messages · Page 1 of 1 (latest)

woeful acorn
hexed egretBOT
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slow river
woeful acorn
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so like i have gotten so far till [1000 950]x^5 by just doing [x+1]^1000 summation

slow river
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Okay, no.

woeful acorn
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now i am trying by summation u+1 x^u 1+x^100-7

slow river
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That's not what you want to do.

woeful acorn
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oh

slow river
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First, represent the sum in sigma notation.

woeful acorn
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like this??

slow river
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Right.

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Now expand (1 + x)^(1000 - u) using the binomial theorem. You'll get a second series.

woeful acorn
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i am getting smtg weird

slow river
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What?

woeful acorn
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lemme try again i am pretty sure i am wrog

slow river
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Show me what you got.

woeful acorn
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i am writing so

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okay

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well

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1000-uCu, x^u

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my interent sucks just a minute

slow river
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Use a different index variable.

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This is a different series.

woeful acorn
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wdym

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like
1000-u
x-u

slow river
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When you're not talking about u.

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Look. Write out the whole double series.

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And then post it.

woeful acorn
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okay

slow river
woeful acorn
slow river
woeful acorn
woeful acorn
slow river
# woeful acorn

This series. Inside this series, expand (1 + x)^(1000 - u).

woeful acorn
slow river
woeful acorn
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OH LIKE THAT

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okay i am being dumb rn
i have never expanded a binomial inside a series like that tbh so i might need help with that

slow river
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It's just going to be a second series, inside the first, with a different index.

shadow grotto
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Try this..

slow river
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I'm already helping.

shadow grotto
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Okay sir

woeful acorn
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how it came 50-u

slow river
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,rotate

toxic sapphireBOT
slow river
woeful acorn
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oh we expand it that

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so now basically should i put n=50?

slow river
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No.

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You have two x terms.

woeful acorn
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hmm

slow river
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Distributive property.

woeful acorn
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we are gonna like multiply them??

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is that possible with sigma???

slow river
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Sigma is just shorthand for a big sum.

woeful acorn
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yea okay so it would be basically
sigma x sigma
([n+1]x^n)*x^k

slow river
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$\sum_{n = 0}^{1000}\sum_{k = 0}^{1000 - n}(n + 1){{1000 - n} \choose k} x^nx^k$

toxic sapphireBOT
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Techie Literate

352737152154468352.png
woeful acorn
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RIGHT

slow river
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Which we can simplify further

woeful acorn
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OMG WAIT SO WE CAN SAY
n+k=50
so
k=n-50 and replace it???

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50-n*

slow river
slow river
# woeful acorn

First, we don't need the series as k varies if k isn't going to vary.

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Second, we only want the coefficient.

woeful acorn
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yea so x^50 doesnt matter now?

slow river
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Third, nCk is undefined for k < 0, so we only need the sum as n goes from 0 to 50.

woeful acorn
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k has a definite value now

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while n varies

slow river
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Right.

woeful acorn
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so thats it??

slow river
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I mean, you still presumably need to evaluate it.

woeful acorn
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we would get some really big number

slow river
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Yeah. WA runs out of time trying to calculate it.

woeful acorn
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ig thats the answer our sir is expecting at 11th grade basics level

slow river
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And it says "coefficients".

woeful acorn
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right

slow river
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So I'm not actually totally sure what it's looking for because obviously there's only one coefficient.

woeful acorn
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i will ask our tr tom and tell you

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TYSMM THO