# A combinatorial identity for $\sum_{k=0}^i \binom{n}{k}$?

Is there a combinatorial identity for the following: $$\sum_{k=0}^{i}\binom{n}{k}$$ for arbitrary integers $n, i$ with $n > i$? If so, what is this identity called?

• Alas, no, except for special values of $i$. – vadim123 Feb 20 '14 at 2:51
• These are values like $n$, $n/2$, yes? – Ayesha Feb 20 '14 at 3:11
• Right, and also silly things like $0,1,2,3$. – vadim123 Feb 20 '14 at 4:33

Using a CAS, I found this "nice" expression $$\sum_{k=0}^{i}\binom{n}{k}=2^n-\binom{n}{i+1} \, _2F_1(1,i-n+1;i+2;-1)$$ May I suggest we name it, at least for the time being, Ayesha's identity.