I need to know if we can derive any formula to calculate this


I don't know if this question has been asked or not (or I didn't search for the correct keyword).

While going through the internet, I found that for $i=0$, the sum would be $2^n$. Also, I came across a equation which I think can be used to derive a formula for the above summation, which was $$\binom{n+1}{r}=\binom{n}{r}+\binom{n}{r-1}$$

  • 2
    $\begingroup$ In order to calculate sum from k to n, you can split $$ \sum_{i = 0}^n = \sum_{i=0}^{k-1} + \underbrace{\sum_{i= k}^n }_{\text{your term}} $$ $\endgroup$ – Matti P. Aug 10 '18 at 10:05
  • $\begingroup$ How will this decrease my calculation time. $\endgroup$ – Punit Jain Aug 10 '18 at 10:26
  • $\begingroup$ There is no closed-form solution to this problem. $\endgroup$ – N. F. Taussig Aug 10 '18 at 11:45
  • $\begingroup$ @N.F.Taussig can you help me with some approach, as i want to compute the result for large numbers. $\endgroup$ – Punit Jain Aug 12 '18 at 5:30

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