I wanted to find the sum of the following series :
$$\sum_{k=1}^{r}{{n}\choose{k}}$$
I searched for it in my book but didn't find it. However I was able to find : $$\sum_{k=1}^{r}{{n}\choose{k}}(-1)^{k}$$
The solution was given as :
When I tried to do the same for the first series I got stuck on : $$2^{n}-\sum_{k=1}^{n-r}{{n-k}\choose{r}}2^{k-1}$$ I am not able to get any ideas related to this.
So how to proceed further with this approach or is there a better method?