Is there a closed formula for the sum of a geometric progression with binomial coefficients?

Question

The title asks it all. $$\sum_{i=0}^n{n\choose i}x^{i+1}=?$$

Answer 1

The binomial theorem gives $$(1+x)^n = \sum_{i=0}^n \binom n i x^i.$$ So your sum in question is simply $x(1+x)^n$.