Telescoping

Write each term as a difference and watch it collapse.

If a term can be written $f(k) - f(k+1)$, the sum collapses to the two ends. The classic is $\frac{1}{k(k+1)} = \frac1k - \frac1{k+1}$, which turns an intimidating sum into a one-line answer.

The same idea drives telescoping products, and it is the discrete cousin of the fundamental theorem of calculus: sum a difference, keep only the boundary.

Whenever a sum has a rigid, repeating shape, spend a moment hunting for the difference form before doing anything harder.

More techniques

• SymmetryIf the problem doesn't care, neither should your answer.
• Working BackwardsStart at the end. The end knows more.
• All 29 techniques →