Geometric sequence formulas (with intuition)

A geometric sequence is one of the most “real-life friendly” ideas in algebra because it models anything that grows or shrinks by a constant factor each step. If something multiplies by the same number over and over, you’re basically living inside a geometric sequence.

The common ratio is the multiplier that takes you from one term to the next: r = a₂ / a₁, and also r = a₃ / a₂, etc. If the sequence really is geometric, every consecutive ratio is the same.

• If r > 1, the sequence grows (3, 6, 12, 24…).
• If 0 < r < 1, it decays toward 0 (100, 50, 25, 12.5…).
• If r = 1, it stays constant (7, 7, 7, 7…).
• If r < 0, it alternates signs (2, −6, 18, −54…).

The nth term formula is a compact way to jump straight to any position in the sequence without listing everything. Why does the exponent use (n − 1)? Because when n = 1, you want a₁—and r^0 = 1. Each time you move one step forward, you multiply by r one more time, which is exactly what exponents represent.

For example, with a₁ = 3 and r = 2, you get: a₁ = 3, a₂ = 3·2, a₃ = 3·2², and so on. So the 8th term is a₈ = 3 · 2⁷ = 384.

Sometimes you don’t just want one term—you want the total of the first n terms: Sₙ = a₁ + a₂ + … + aₙ. For geometric sequences, there’s a classic cancellation trick that gives a closed-form sum.

Start with Sₙ = a₁ + a₁r + a₁r² + … + a₁r^(n−1). Multiply both sides by r: rSₙ = a₁r + a₁r² + … + a₁rⁿ. Subtract: Sₙ − rSₙ = a₁ − a₁rⁿ. Factor: Sₙ(1 − r) = a₁(1 − rⁿ). Divide: Sₙ = a₁(1 − rⁿ)/(1 − r) (for r ≠ 1).

If r = 1, the sequence is constant, so Sₙ = n·a₁.

If |r| < 1, the terms shrink and the total can approach a finite value even with infinitely many terms. In that case: S∞ = a₁ / (1 − r).

Example: 1 + 1/2 + 1/4 + 1/8 + … has a₁ = 1, r = 1/2, so S∞ = 1/(1 − 1/2) = 2. This is also how repeating decimals become fractions.

Examples you can copy for homework

The fastest way to get confident is to run a few “template problems” and notice the same patterns repeating.

Sequence: 5, 15, 45, 135, … has a₁ = 5, r = 3. So a₁₀ = 5 · 3^(10−1) = 98415.

With a₁ = 5, r = 3, n = 6: S₆ = 5(1 − 3⁶)/(1 − 3) = 1820.

Sequence: 2, −6, 18, −54, … has r = −3. The 7th term is a₇ = 2 · (−3)⁶ = 1458 (positive because the exponent is even).

Sequence: 80, 40, 20, 10, … has r = 1/2. Since |r| < 1, S∞ = 80/(1 − 1/2) = 160.

If a₁ = 3, a₈ = 384, then 384 = 3 r⁷ → r⁷ = 128 → r = 2.

Viral trick: compare r = 1.05 vs r = 1.10 over many steps. Tiny changes compound dramatically.

What this calculator is doing under the hood

This page offers multiple “solve modes” because real problems don’t always give you the same variables.

• nth term: computes aₙ = a₁ r^(n−1).
• finite sum: computes Sₙ (or n·a₁ when r=1).
• infinite sum: computes S∞ = a₁/(1−r) only when |r| < 1.
• find r: uses r = (aₙ/a₁)^(1/(n−1)).
• find a₁: uses a₁ = aₙ / r^(n−1).

• r = 1: sum becomes Sₙ = n·a₁.
• |r| ≥ 1: infinite sum is flagged as non-convergent.
• n must be integer ≥ 1: because “term number” is a position.
• Negative ratio in “find r”: real roots depend on parity; the tool warns when no real r exists.

Frequently Asked Questions

• Arithmetic vs geometric—what’s the difference?

  Arithmetic adds a constant difference each step. Geometric multiplies by a constant ratio each step. “Doubling/halving/compounding” screams geometric.

• Why is it r^(n−1)?

  Because when n=1 you need a₁, and r^0 = 1. Each step forward adds one more multiplication by r.

• When does an infinite sum exist?

  Only when |r| < 1. Otherwise the terms don’t shrink fast enough (or at all) so the sum diverges.

• Can r be negative?

  Yes—then terms alternate signs. The magnitude grows or shrinks based on |r|.

• How do I find r from a₁ and aₙ?

  Rearrange: r = (aₙ/a₁)^(1/(n−1)). That’s the “common ratio” mode here.

• Where is this used in real life?

  Compound interest, depreciation, population change, repeated discounts, and any “multiply by a percent each period” scenario.

Make this shareable (without changing the math)

Geometric sequences go viral because small differences in r explode over time. Try side-by-side scenarios and screenshot the result box (it includes substituted steps).

• Try a₁ = 1, n = 365 and compare r = 1.01 vs r = 0.99.
• Try discounts: start at $100, apply r = 0.8 repeatedly.
• Try “followers”: start at 200, test r = 1.05 for 52 weeks.
• Copy the result text and challenge a friend to beat your number with a different r.