Geometric Sequence

A sequence where each term is found by multiplying the previous term by a constant (common ratio).

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A sequence where each term is found by multiplying the previous term by a constant (common ratio).

A geometric sequence (also called geometric progression) 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, compound interest, population growth, and many natural phenomena.

aₙ = a₁ × rⁿ⁻¹, where r is the common ratio

2, 6, 18, 54, 162... (common ratio = ×3)
1000, 500, 250, 125... (common ratio = ×0.5 or ÷2)
1, -2, 4, -8, 16... (common ratio = ×(-2))

How to recognize it

Divide each term by the previous term
If all ratios are the same, it's geometric
The pattern is 'multiply by the same number each time'

Common mistakes

Confusing with arithmetic sequences (add vs multiply)
Forgetting that division is multiplication by a fraction
Missing negative ratios that cause alternating signs

Step-by-step walkthrough

Find the next term: 2, 6, 18, 54, ?

Calculate ratios: 6÷2=3, 18÷6=3, 54÷18=3
All ratios are the same: common ratio = 3
This is geometric. Multiply the last term by 3
The next term is 162

Answer: 162

Practice guidance

Number Sequences