How do u prove $\sum_{k=0}^n\frac{1}{2^{k}}\binom{n+k}{n} = 2^{n}$?

Asked Nov 25 '20 at 11:04

Active Nov 25 '20 at 11:06

Viewed 67 times

Here is the expression

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

How to go about proving this? I tried expanding the expression and it seems that $n+1$ must be even. But it isn't helping much. Can someone please help? Can this be proved using generating functions or combinatorics?

edited Nov 25 '20 at 11:06

Sewer Keeper

1,440

asked Nov 25 '20 at 11:04

nmnsharma007

1

Yes,Thank you. I couldn't find it on my own. Should I delete my question now or let it remain? – nmnsharma007 Nov 25 '20 at 11:14
1

I believe either option is fine. I found this with Approach0 (https://approach0.xyz/search/), and it has proven to be quite powerful. – player3236 Nov 25 '20 at 11:17
1

WOW!!. I am gonna let my question be here. Someone like me in the future will surely want to know the link u gave.Thanks man!!! – nmnsharma007 Nov 25 '20 at 11:19

How do u prove $\sum_{k=0}^n\frac{1}{2^{k}}\binom{n+k}{n} = 2^{n}$?

0 Answers0