Telescoping Sum

A sum in which subsequent terms cancel each other, leaving only initial and final terms. For example,

$$\begin{eqnarray*} S&=&\sum_{i=1}^{n-1} (a_i-a_{i+1})\\ &=&(a_1-a_2)+(a_2-a_3)+\ldots+(a_{n-2}-a_{n-1})+(a_{n-1}-a_n)\\ &=&(a_1-a_n) \end{eqnarray*}$$

is a telescoping sum.

See also Zeilberger's Algorithm