Geometric Sequence Formulas
Find the nth term and sum of a geometric sequence.
Formulas: aₙ = a₁ × r^(n-1) and S = a₁(1-rⁿ)/(1-r) with worked examples.
The Formulas
nth term: aₙ = a₁ × r^(n-1)
Sum of n terms: S = a₁ × (1 - rⁿ) / (1 - r) (when r ≠ 1)
A geometric sequence is a list of numbers where each term is multiplied by the same fixed amount.
That fixed amount is called the common ratio.
Variables
| Symbol | Meaning |
|---|---|
| aₙ | The nth term in the sequence |
| a₁ | The first term in the sequence |
| n | The position of the term |
| r | The common ratio between consecutive terms |
| S | The sum of the first n terms |
Example 1
Find the 8th term of the sequence: 3, 6, 12, 24, ...
a₁ = 3, r = 6 / 3 = 2, n = 8
aₙ = a₁ × r^(n-1)
a₈ = 3 × 2^(8-1) = 3 × 2⁷ = 3 × 128
a₈ = 384
Example 2
Find the sum of the first 5 terms of: 4, 12, 36, 108, ...
a₁ = 4, r = 12 / 4 = 3, n = 5
S = a₁ × (1 - rⁿ) / (1 - r)
S = 4 × (1 - 3⁵) / (1 - 3) = 4 × (1 - 243) / (-2)
S = 4 × (-242) / (-2) = 4 × 121
S = 484
When to Use It
Use geometric sequence formulas when:
- Each term is a fixed multiple of the previous one (doubling, tripling, halving, etc.)
- Calculating compound interest or exponential growth
- Working with population growth or radioactive decay
- Solving problems involving repeated percentage increases or decreases