A geometric sequence is a list of numbers where each term is the previous term multiplied by a fixed number called the common ratio (r). A geometric series is the SUM of those terms. The two formulas you need are simple, but knowing when to use which — and when the infinite version converges — is what trips students up.
aₙ = a × rⁿ⁻¹
Where a is the first term, r is the common ratio, n is the term you want (1, 2, 3, …).
Example: in 2, 6, 18, 54, … a = 2 and r = 3, so a₅ = 2 × 3⁴ = 2 × 81 = 162.
Sₙ = a × (1 − rⁿ) / (1 − r), valid for r ≠ 1
If r = 1, every term equals a, so just multiply: Sₙ = n × a.
Example: sum of 5, 10, 20, 40, 80 (a = 5, r = 2, n = 5):
S₅ = 5 × (1 − 2⁵) / (1 − 2) = 5 × (1 − 32) / (−1) = 5 × (−31) / (−1) = 155
S∞ = a / (1 − r), valid only when |r| < 1
Example: 1 + ½ + ¼ + ⅛ + … (a = 1, r = ½, |r| < 1 ✓)
S∞ = 1 / (1 − ½) = 1 / 0.5 = 2.
If |r| ≥ 1, terms either stay constant or grow without bound, so the infinite sum is ∞ (diverges).
| Question type | Use this formula |
|---|---|
| “What is the 12th term of 3, 9, 27, …?” | aₙ = a × rⁿ⁻¹ → a₁₂ = 3 × 3¹¹ |
| “What is the sum of the first 10 terms of 2, 4, 8, 16, …?” | Sₙ = a(1 − rⁿ)/(1 − r) → S₁₀ = 2(1 − 2¹⁰)/(1 − 2) = 2046 |
| “What is 0.999… as a fraction?” (geometric) | S∞ = a/(1 − r) → 0.9/(1 − 0.1) = 0.9/0.9 = 1 |
| “Does the series 1 + 2 + 4 + 8 + … converge?” | r = 2, |r| ≥ 1, so it diverges (sum is ∞) |
a = 5, r = 15/5 = 3, n = 8.
a₈ = 5 × 3⁷ = 5 × 2187 = 10,935
a = 100, r = ½, n = 6.
S₆ = 100 × (1 − (½)⁶) / (1 − ½)
(½)⁶ = 1/64
S₆ = 100 × (63/64) / (½) = 100 × (63/64) × 2 = 196.875
a = 4, r = ⅓, |r| < 1 ✓
S∞ = 4 / (1 − ⅓) = 4 / (⅔) = 6
a₅/a₁ = r⁴ → 48/3 = r⁴ → r⁴ = 16 → r = ±2 (both work)
Sₙ = 2(1 − 3ⁿ)/(1 − 3) = (3ⁿ − 1) ≥ 1000
3ⁿ ≥ 1001 → n × log(3) ≥ log(1001) → n ≥ log(1001)/log(3) ≈ 6.29
So n = 7 terms. Check: S₇ = (3⁷ − 1) = 2187 − 1 = 2186 ✓
For a one-click way to compute any of these, try our Geometric Sequence Calculator — enter a, r, n and it returns nth term, partial sum, and (if applicable) the infinite sum.
Is “geometric series” the same as “geometric sequence”? The sequence is the LIST of terms (2, 6, 18). The series is the SUM of those terms (2 + 6 + 18 = 26). Same numbers, different operation.
Why is it called “geometric”? Because the geometric mean of two terms equals the term between them. In 2, 6, 18, the middle term 6 = √(2 × 18). Compare to arithmetic where the middle term is the average.
Real-world examples? Compound interest (each year multiplies by 1 + rate), bouncing balls (each bounce reaches a fixed fraction of previous height), radioactive decay, population growth. Anything that scales by a constant multiplier per step.