Geometric Sequences

A geometric sequence (also called a geometric progression, or GP) is a sequence where each term is obtained by multiplying the previous term by a fixed number called the common ratio. Geometric sequences model exponential growth and decay and are widely tested on the Edexcel A-Level (9MA0).

---

Key Definitions

Term	Meaning
Geometric sequence	A sequence with a constant ratio between consecutive terms
Common ratio (r)	The multiplier: r = u(n+1) / u(n)
First term (a)	The starting value, u(1)
Convergent GP	A GP where

---

The nth Term Formula

For a geometric sequence with first term a and common ratio r:

u(n) = a x r^(n-1)

Worked Example 1

Find the 8th term of the GP 3, 6, 12, 24, ...

• a = 3, r = 6/3 = 2
• u(8) = 3 x 2^7 = 3 x 128 = 384

Worked Example 2

The 2nd term of a GP is 12 and the 5th term is 324. Find a and r.