Question

Binomial Theorem, finding the nth term

Answer 1

The questions says Find the specified term of each binomial expansion.
I need the find the twelfth term of (2+x)^11

Answer 2

I did it and got it

Answer 3

I just don't know which term is the one its talking about

Answer 4

does the twelfth term mean the first or last one?

Answer 5

I'd say last one. What did you get as your answer?

Answer 6

I got 2048

Answer 7

It seems like that would be the first term.

Answer 8

Oh so x^11 is the twelfth term?

Answer 9

Yeah, because it is (2+x), the last term would be x^11. If it was (x+2) than the last term would be the 2048.

Answer 10

I need more help with this. Could you help me

Answer 11

@oldrin.bataku

Answer 12

yes

Answer 13

@jollyjolly0 so does the order of the equation change everything?

Answer 14

The binomial theorem comes from
$$
(x+y)^n=\Sigma_{k=0}^n \binom{n}{k}x^ky^{n-k}
$$

So, the kth term would be
$$
\binom{n}{k}x^ky^{n-k}
$$
In your case k=11, y=2.
So, for k=11, which is the 12th term (because we start with zero):
$$
\binom{11}{11}x^{11}2^{11-11}\\
=x^{11}
$$

Answer 15

could you teach me how to do it? I want to learn to do this for my upcoming test

Answer 16

Yes, look at the pascals triangle (i attached one but you can just google it if you want).

the nth term of the expanded binomial (a+b)^p is:
\[c*a ^{P+1-n}*b ^{n-1}\]


in order to find c:

look at the (p+1)th row on the pascal's triangle. then take the nth number from the left on that row. that number is c

Answer 17

It turns out that when you multiply out \((x+y)^n\), there will be \(\binom{n}{k}\) terms where x has power terms \(x^k\) and y has power terms \(y^{n-k}\). This nice relationship is determined by calculating the number of ways to make a group of size k from n objects.