How to simplify this summation: $\frac{2\sum_{k=0}^n2^n}{n+1}=2^{n+1}$?

Question

So I saw an earlier post where they had this equation here.

$\frac{2\sum_{k=0}^n2^n}{n+1}=2^{n+1}$?

However, I do not know how they did this? Am I missing something?

The sum is $\underbrace{2^n+2^n+\cdots+2^n}_{n+1\text{-terms}}\ $; it simplifies to $(n+1) 2^n$. — David Mitra, Dec 14 '14 at 15:03
$n$ is arbitrary; there is no $k$ anywhere in the expression, so everything can be factored outside of the summation sign. — Sultan of Swing, Dec 14 '14 at 15:14

score 3 · Answer 1 · answered Dec 14 '14 at 15:06

3

Hint: $\sum_{k=0}^n 2^n=2^n\cdot (n+1)$ Note, that $2^n$ have no index k.

answered Dec 14 '14 at 15:06

callculus42

score 3 · Accepted Answer · answered Dec 14 '14 at 15:07

3

If the expression is correct $$\frac{2\sum_{k=0}^n2^n}{n+1}=\frac{2\times 2^n}{n+1}\sum_{k=0}^n1=\frac{2^{n+1}}{n+1}(n+1)=2^{n+1}$$

answered Dec 14 '14 at 15:07

Claude Leibovici

I always been slow when it came to summations. – Chinedum Ukejianya Dec 14 '14 at 15:19
You are very welcome ! If every single time I have been slow, I was given a cent, I should be a millionaire. So, don't worry. Cheers :-) – Claude Leibovici Dec 14 '14 at 15:21

2 Answers2