What is a geometric progression?
A geometric progression (GP) is a sequence in which each term is the previous term multiplied by a fixed number called the common ratio r. For example 3, 6, 12, 24, … has a = 3 and r = 2.
Key formula
nth term: Tn = a rn−1
Sum of n terms: Sn = a(rn − 1)/(r − 1), r ≠ 1
Sum to infinity: S∞ = a/(1 − r), valid only when |r| < 1.
Finding a term and the ratio
The common ratio is any term divided by the one before it: r = T2 ÷ T1. Then the nth-term formula gives any term.
- For 5, 10, 20, …: r = 2, so T5 = 5 × 24 = 80.
- If T2 = 6 and T4 = 54, then r2 = 54/6 = 9, so r = 3.
Worked example
A GP has a = 3 and r = 2. Find T4 and S5.
T4 = 3 × 23 = 3 × 8 = 24.
S5 = 3(25 − 1)/(2 − 1) = 3(32 − 1)/1 = 93.
Sum to infinity
When |r| < 1 the terms shrink towards zero and the sum settles on a finite value. For 8 + 4 + 2 + …, a = 8 and r = 1/2, so S∞ = 8/(1 − 1/2) = 16.
Remember
- S∞ exists only when −1 < r < 1.
- The power in Tn is rn−1, not rn.
- r can be a fraction or negative.