Is this a correct way to rewrite the binomial theorem?

1. Mar 10, 2012

jey1234

1. The problem statement, all variables and given/known data
I am doing a poof and I need to use the binomial theorem. However is the following a correct way to rewrite it?

$$(a+b)^n\ =\ {n \choose 0}a^{n} + \sum_{k=1}^{n}{n \choose k}\ a^{n-k}\ b^{k}$$

2. Relevant equations

$$(a+b)^n\ =\ \sum_{k=0}^{n}{n \choose k}\ a^{n-k}\ b^{k}$$

3. The attempt at a solution
Basically, I want to extract the first term of the binomial expansion out of the summation but I'm not that good with sigma notation. Don't I just have to increase the lower bound by 1 and write the first term outside (as shown above)? Thanks.

2. Mar 10, 2012

tiny-tim

hey jey1234!

yes, that's fine

3. Mar 10, 2012

jey1234

thanks tim. one more quick question. is the following correct?

$$\frac{1}{a}\sum_{k=1}^{n}a^{k}\ b^{k}\ =\ \sum_{k=1}^{n}a^{k-1}\ b^{k}$$

4. Mar 10, 2012

SammyS

Staff Emeritus
Yes, that's correct.

5. Mar 10, 2012

thanks sammy