Geometric Sequence Calculator — nth Term, Sum & Infinite Series
Calculate geometric sequence terms and sums. Covers nth term formula, finite and infinite series sums, common ratio, and the compound interest connection with worked examples.
In a geometric sequence, each term is multiplied by the same ratio to get the next one. Compound interest, population growth, radioactive decay, and the area of a fractal — they're all geometric progressions. Find any term or sum with the CalcHub Geometric Sequence Calculator.
Key Definitions
First term (a₁): The initial value. Common ratio (r): The constant multiplier between terms. Can be positive, negative, or fractional. nth term (aₙ): Value at position n.A sequence is geometric if the ratio between consecutive terms is constant:
aₙ₊₁/aₙ = r for all n
The nth Term Formula
$$a_n = a_1 \cdot r^{n-1}$$
Example: Find the 8th term of 3, 6, 12, 24, ...Here a₁ = 3, r = 2.
a₈ = 3 × 2⁷ = 3 × 128 = 384
r = 0.5
a₆ = 100 × (0.5)⁵ = 100 × 1/32 = 3.125
Sum of Finite Geometric Series
$$S_n = a_1 \cdot \frac{1 - r^n}{1 - r} \quad (r \neq 1)$$
If r = 1: Sₙ = n × a₁ (constant sequence)
Example: Sum of first 6 terms of 3, 6, 12, 24, 48, 96S₆ = 3 × (1 − 2⁶)/(1 − 2) = 3 × (1 − 64)/(−1) = 3 × 63 = 189
Quick check: 3+6+12+24+48+96 = 189 ✓
Sum of Infinite Geometric Series
When |r| < 1, the series converges to:
$$S_\infty = \frac{a_1}{1 - r}$$
When |r| ≥ 1, the infinite sum doesn't exist (diverges).
Example: Sum of 1 + 1/2 + 1/4 + 1/8 + ...a₁ = 1, r = 0.5
S∞ = 1/(1 − 0.5) = 1/0.5 = 2
This makes physical sense: Zeno's paradox involves an infinite number of steps that sum to a finite total.
Example: 0.333... as a fraction 0.333... = 3/10 + 3/100 + 3/1000 + ... a₁ = 0.3, r = 0.1 S∞ = 0.3/(1 − 0.1) = 0.3/0.9 = 1/3 ✓Common Ratio Reference
| r Value | Behavior |
|---|---|
| r > 1 | Terms grow (exponential growth) |
| r = 1 | Constant sequence |
| 0 < r < 1 | Terms shrink toward 0 (converges) |
| r = 0 | All terms after a₁ are 0 |
| −1 < r < 0 | Alternating, shrinking (converges) |
| r = −1 | Alternating ±a₁ (diverges) |
| r < −1 | Alternating, growing (diverges) |
Compound Interest Connection
Compound interest is a geometric sequence in disguise.
Balance after n years: A = P × (1 + i)ⁿ⁻¹ (when P is the first term)
More precisely:
- Initial: P
- After 1 year: P(1+i)
- After 2 years: P(1+i)²
- After n years: P(1+i)ⁿ
This is aₙ = a₁ × rⁿ with a₁ = P, r = (1+i), and "n" being the time period. Example: ₹10,000 at 8% per year for 10 years: A = 10,000 × (1.08)¹⁰ = 10,000 × 2.1589 = ₹21,589
The sequence 10,000; 10,800; 11,664; 12,597; ... is geometric with r = 1.08.
Why does the infinite sum formula only work for |r| < 1?
If |r| ≥ 1, each term is at least as large as the previous one — the terms don't shrink toward zero. For a sum to converge, you need the terms to eventually contribute less and less. The proof uses the formula Sₙ = a₁(1−rⁿ)/(1−r) and asks what happens as n→∞: if |r| < 1, rⁿ→0 and Sₙ→a₁/(1−r). If |r| ≥ 1, rⁿ doesn't go to zero, so the sum grows without bound.
How do I find the common ratio?
Divide any term by the one before it: r = aₙ/aₙ₋₁. If this ratio is the same for every consecutive pair, the sequence is geometric. If you only know two non-consecutive terms aₘ and aₙ, then r^(n−m) = aₙ/aₘ, so r = (aₙ/aₘ)^(1/(n−m)).
What is exponential growth vs. geometric sequence?
They're the same idea in different settings. A geometric sequence is discrete (defined at whole number positions). Exponential growth is continuous (defined for all real values of time). The formula A = Peʳᵗ is the continuous version of the geometric sequence formula.