Learn on PengiAoPS: Introduction to Algebra (AMC 8 & 10)Chapter 21: Sequences & Series

Lesson 5: Telescoping

In this Grade 4 AMC Math lesson from AoPS Introduction to Algebra, students learn the telescoping technique for simplifying sums and products by identifying and canceling consecutive terms that appear with opposite signs or as factors in numerators and denominators. Working through problems in Chapter 21, students apply telescoping to expressions involving integer differences, radical denominators requiring rationalization, and fraction products to reduce complex multi-term expressions down to just a first and last term. The lesson also introduces partial fraction decomposition as a strategy for rewriting terms so that a series telescopes.

Section 1

Telescoping Series Fundamentals

Property

A telescoping series is a sum where consecutive terms cancel out, leaving only the first and last terms:

∑k=1n(ak−ak+1)=a1−an+1\sum_{k=1}^{n} (a_k - a_{k+1}) = a_1 - a_{n+1}

Examples

Section 2

Identify Telescoping Patterns in Partial Sums

Property

A telescoping series has partial sums SnS_n that follow a predictable pattern where most intermediate terms cancel, leaving only the first and last terms: Sn=a1−an+1S_n = a_1 - a_{n+1} or similar simplified form.

Examples

Section 3

Partial Fraction Decomposition for Telescoping

Property

Common telescoping partial fraction formulas:

1k(k+1)=1k−1k+1\frac{1}{k(k+1)} = \frac{1}{k} - \frac{1}{k+1}
1k(k+2)=12(1k−1k+2)\frac{1}{k(k+2)} = \frac{1}{2}\left(\frac{1}{k} - \frac{1}{k+2}\right)
1(k+a)(k+b)=1b−a(1k+a−1k+b)\frac{1}{(k+a)(k+b)} = \frac{1}{b-a}\left(\frac{1}{k+a} - \frac{1}{k+b}\right)

Examples

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Telescoping Series Fundamentals

Property

A telescoping series is a sum where consecutive terms cancel out, leaving only the first and last terms:

∑k=1n(ak−ak+1)=a1−an+1\sum_{k=1}^{n} (a_k - a_{k+1}) = a_1 - a_{n+1}

Examples

Section 2

Identify Telescoping Patterns in Partial Sums

Property

A telescoping series has partial sums SnS_n that follow a predictable pattern where most intermediate terms cancel, leaving only the first and last terms: Sn=a1−an+1S_n = a_1 - a_{n+1} or similar simplified form.

Examples

Section 3

Partial Fraction Decomposition for Telescoping

Property

Common telescoping partial fraction formulas:

1k(k+1)=1k−1k+1\frac{1}{k(k+1)} = \frac{1}{k} - \frac{1}{k+1}
1k(k+2)=12(1k−1k+2)\frac{1}{k(k+2)} = \frac{1}{2}\left(\frac{1}{k} - \frac{1}{k+2}\right)
1(k+a)(k+b)=1b−a(1k+a−1k+b)\frac{1}{(k+a)(k+b)} = \frac{1}{b-a}\left(\frac{1}{k+a} - \frac{1}{k+b}\right)

Examples