Geometric Progression Cheat Sheet

Geometric Progression Cheat Sheet

Quick Revision

A geometric progression (GP) is a sequence in which each term is a fixed multiple of the one before it, the common ratio r. With first term a, the terms are a, ar, ar², ar³, ….

1. The nth Term

aₙ = a × r⁽ⁿ⁻¹⁾

Here a is the first term, r the common ratio and n the term number.

2. Sum of the First n Terms

Sₙ = a(rⁿ − 1) / (r − 1)      (r > 1)
Sₙ = a(1 − rⁿ) / (1 − r)      (r < 1)

The two forms are the same equation; pick the one that keeps the numerator positive.

3. Sum to Infinity

For a GP whose terms shrink — that is, |r| < 1 — the sum of all its terms is finite:

S∞ = a / (1 − r)

4. Geometric Mean

The geometric mean of two positive numbers a and b is:

GM = √(a × b)

If three numbers are in GP, the middle one is the GM of the other two.

5. Worked Example

For the GP 2, 6, 18, 54, … the first term is a = 2 and r = 3.

5th term   = 2 × 3⁴ = 2 × 81 = 162
Sum of 5   = 2(3⁵ − 1) / (3 − 1) = 2 × 242 / 2 = 242

For 1 + 1/2 + 1/4 + … (here r = 1/2): S∞ = 1 / (1 − 1/2) = 2.

6. Exam Quick Tips

• The common ratio is any term ÷ its previous term; it stays the same throughout.
• The sum to infinity exists only when |r| < 1.
• Three numbers in GP are best written as a/r, a, ar — their product is a³.

Keep revising with the rest of the Maths section:

• All Maths topics
• Arithmetic Progression Cheat Sheet
• Logarithms Cheat Sheet

How this was made: Drafted with AI assistance; spot-checked by the editor on 31 October 2026.