Rewriting summation formula I just watched a tutorial on recurrence by substitution. In the tutorial, it mentioned about rewriting
$\sum\limits_{i=1}^\mathbb{k}{2^i}$
as (2k+1 - 2). My question is can I generalize it as xlimit + 1 - x where x is the base. 

Let $S = \sum_{i=1}^k x^i$. Consider the following manipulation $$S = x + x^2 + x^3 + \cdots + x^k = x(1 + x + \cdots + x^{k-1}) = x(S + 1 - x^k)$$
It follows that $S = (x^{k+1} - x)/(x - 1)$
