Is this a correct way to rewrite the binomial theorem?

[jey1234]

Homework Statement

I am doing a poof and I need to use the binomial theorem. However is the following a correct way to rewrite it?

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

Homework Equations

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

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.

 

[tiny-tim]

hey jey1234!

yes, that's fine

 

[jey1234]

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

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

 

[SammyS]

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

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

Yes, that's correct.

 

[jey1234]

thanks sammy