Question

the coefficient of x^2 in the expansion of (1 + x/5)^n, where n is a positive integer, is 3/5. Find the value of n.

How to do this???? :P

Answer 1

\[(1+x)^n=1+nx+\dbinom{n}{2}x^2+...\] in your case you will have
\[1+n\frac{x}{5}+\dbinom{n}{2}\frac{x^2}{25}+ ...\]

Answer 2

so basically i need to multiply 1/25 and \[\left(\begin{matrix}n \\ 2\end{matrix}\right)\], but how do I do that?

Answer 3

and equate it to 3/5.

Answer 4

so you know that
\[\frac{n(n-1)}{2}\times \frac{1}{25}=\frac{3}{5}\]
\[\frac{n(n-1)}{2\times 25}=\frac{3}{5}\]
\[\frac{n(n-1)}{10}=3\]
\[n(n-1)=30\]

Answer 5

and since n is a whole number, only possible answer is
\[n=6\]

Answer 6

wait, sorry, I don't get why n(n - 1)/2

Answer 7

oh ok lets go slower

Answer 8

to compute
\[\dbinom{n}{k}\] you do not really use
\[\frac{n!}{k!(n-k)!}\] because it it is too much work, you just cancel. but we can write it out anyway

Answer 9

\[\dbinom{n}{2}=\frac{n!}{2!(n-2)!}\] ok?

Answer 10

ohhh. so if it was the coefficient of x^3, then we would do n(n - 1)(n - 3)/3 ??

Answer 11

then numerator is
\[n\times (n-1)\times (n-2)!\] so you can cancel the
\[(n-2)!\] and get
\[\frac{n(n-1)}{2}\]

Answer 12

it would be
\[\frac{n(n-1)(n-2)}{3!}\]

Answer 13

don't forget the factorial in the denominator

Answer 14

^sorry, that's what I meant to type :P Got it.

Answer 15

great!

Answer 16

So, what's up, satellite??? haven't been here in a while :P

Answer 17

not much. where you been?

Answer 18

Sitting at home, studying for my mocks -_-. Ohhh, and i've got braces. i look WEIRD.

Answer 19

been a month it seems

Answer 20

ouch. did they hurt?

Answer 21

well, i know. ^nopee, not much, but after a few hours they're tickling me :P And I feel like a kid, 'cause over here most people had their braces off by the 7th / 9th grade, and i'm starting in the 10th.

Answer 22

But i've got a pretty blue, AND it's only for a year, so it's cool.

Answer 23

well at least they are a color and not those awful railroad tracks!

Answer 24

*7th/8th grade. I KNOW, those are horrid. My brother didn't put any colours - I killed him for it. Idiot.

Answer 25

More than that, I know some people who have gotton the clear, translucent braces, not the metal ones, and they've put colours on them. Kinda kills the whole point of see-through ones, doesn't it???

Answer 26

the new thing seems to be to make you have them twice. once around 6th /7th, then a year off, then back again. i think it is just a way to get more money out of parents

Answer 27

What cheapsters.

Answer 28

See??/ i'm such a nice girl, I became your fan :P