Telescoping Sums

A telescoping sum is one where most terms cancel out, leaving only the first and last. Consider $\sum_{i=1}^{n} (a_i - a_{i-1})$.

Expand it: $(a_1 - a_0) + (a_2 - a_1) + (a_3 - a_2) + ... + (a_n - a_{n-1})$. Notice $a_1$ cancels with $-a_1$, $a_2$ cancels with $-a_2$, and so on.

What remains is $a_n - a_0$. The sum collapses to just two terms. This pattern appears when computing differences or analyzing algorithms that maintain running totals.