The question is whether the below is true.

Sum[Binomial[n, k], {k, 0, s - 1}] == Sum[2^(k - 1)*Binomial[n - k, s - k], {k, 1, s}]

Mathematica can simplify as follows, but it fails to Reduce[] or Solve[].

2^n == Binomial[n, s] Hypergeometric2F1[1, s - n, s + 1, -1]

• Binomial[n - 1, s - 1] Hypergeometric2F1[1, 1 - s, 1 - n, 2]

The question is whether the below is true.

$$\sum _{k=0}^{s-1} \binom{n}{k}=\sum _{k=1}^s 2^{k-1} \binom{n-k}{s-k}$$

Mathematica can simplify as follows, but it fails to Reduce[] or Solve[].

$$2^n=\binom{n}{s} \, _2F_1(1,s-n;s+1;-1)+\binom{n-1}{s-1} \, _2F_1(1,1-s;1-n;2)$$