Geometric Sequence Calculator
Find the nth term, partial sum, and terms of any geometric sequence.
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What Is a Geometric Sequence?
A geometric sequence is a sequence where each term is found by multiplying the previous term by a constant ratio (r). Example: 2, 6, 18, 54, 162... — the common ratio is 3. If |r| > 1 the sequence grows; if |r| < 1 it converges toward zero; if r < 0 it alternates sign.
Formula for the nth Term
aₙ = a₁ × r^(n−1). For 2, 6, 18...: a₅ = 2 × 3⁴ = 2 × 81 = 162.
Sum of n Terms (Partial Sum)
Sₙ = a₁ × (1 − rⁿ) / (1 − r) when r ≠ 1. For r = 1: Sₙ = n × a₁. For 2, 6, 18... with n=5: S₅ = 2 × (1−3⁵)/(1−3) = 2 × (1−243)/(−2) = 2 × 121 = 242.
Infinite Geometric Series
When |r| < 1, the infinite sum converges: S∞ = a₁ / (1 − r). For a₁=1, r=0.5: S∞ = 1/(1−0.5) = 2. This is used in zeno's paradox, physics (bouncing ball), and finance (perpetuities).
Real-World Applications
Compound interest is geometric: each year's balance = previous × (1+rate). Population growth, radioactive decay, depreciation by a percentage, and many natural phenomena follow geometric patterns.
Worked Example: Investment Growth
You invest $1,000 at 8% annual interest. Each year's balance is the previous year multiplied by 1.08, making this a geometric sequence with a₁ = 1000 and r = 1.08.
Year 5: a₅ = 1000 × 1.08⁴ = 1000 × 1.3605 = $1,360.49
Year 10: a₁₀ = 1000 × 1.08⁹ = 1000 × 1.9990 = $1,999.00
Total after 10 years (sum): S₁₀ = 1000 × (1 − 1.08¹⁰) / (1 − 1.08) = approximately $14,487. This represents the total of all annual balances added together.
Geometric vs. Arithmetic Sequences
| Feature | Arithmetic | Geometric |
|---|---|---|
| Pattern | Add constant (d) | Multiply constant (r) |
| nth term | a₁ + (n−1)d | a₁ × r^(n−1) |
| Example | 2, 5, 8, 11... (d=3) | 2, 6, 18, 54... (r=3) |
| Growth type | Linear | Exponential |
| Infinite sum | Diverges always | Converges if |r| < 1 |
Frequently Asked Questions
A sequence where each term is the previous term multiplied by a constant ratio r. Example: 3, 6, 12, 24... (r=2).
aₙ = a₁ × r^(n−1). For a₁=3, r=2: a₅ = 3 × 2⁴ = 48.
Sₙ = a₁(1−rⁿ)/(1−r) for r≠1. For a₁=1, r=2, n=5: S₅ = 1×(1−32)/(−1) = 31.
All terms are equal to a₁ and the sum is n×a₁.
When |r|<1, the infinite sum = a₁/(1−r). Example: 1 + 0.5 + 0.25 + ... = 1/(1−0.5) = 2.
When r is negative, the sequence alternates between positive and negative values. For example, a₁=2, r=−3 gives 2, −6, 18, −54, 162... The partial sums oscillate. If |r| < 1 (e.g., r=−0.5), the infinite series still converges to a₁/(1−r).
Use r = (aₙ/aₘ)^(1/(n−m)). For example, if a₂=6 and a₅=48, then r = (48/6)^(1/(5−2)) = 8^(1/3) = 2. Verify: a₂=6, a₃=12, a₄=24, a₅=48.
A geometric sequence is the discrete version of exponential growth — values are defined only at integer positions. Exponential functions like y = a·rˣ are the continuous counterpart. Compound interest with annual compounding follows a geometric sequence, while continuous compounding uses the exponential function e^(rt).