Olympiad problem-solving technique

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.

Where it appears on Lemma: Level 1 (fractions) and Level 7 (recurrences).

Train it on Lemma

Telescoping unlocks at Level 2 of Lemma's eight-level ladder, with lessons that teach it and drills that make it stick. 78 lessons, 624 curated problems and unlimited generated practice at six difficulties. Free to start — no card, no trial clock.

Find your level

More techniques

SymmetryIf the problem doesn't care, neither should your answer.
Working BackwardsStart at the end. The end knows more.
All 26 techniques →