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You can type decimals or fractions like -3/4. We’ll keep results exact when possible.

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How to use this calculator

  1. Choose what you want to compute (term, sum, infinite sum, or ratio).
  2. Enter the required values (a1, r, n, or two terms).
  3. Click Calculate to get the answer plus (optional) steps, table, and mini visual.
  4. Optional: keep Prefer exact fractions on to avoid rounding issues.

Tip: Infinite sums only exist when |r| < 1.

How this calculator works

  • Term formula: an = a1 rn−1
  • Finite sum: Sn = a1 (1 − rn) / (1 − r) (if r ≠ 1), otherwise Sn = n a1
  • Infinite sum (convergent only): S∞ = a1 / (1 − r) for |r| < 1
  • Ratio from two terms: if ak₁ ≠ 0, then rk₂−k₁ = ak₂ / ak₁

Formula & Equation Used

Geometric term: an = a1 rn−1

Finite geometric sum: Sn = a1 (1 − rn) / (1 − r) (for r ≠ 1)

Infinite geometric sum: S∞ = a1 / (1 − r) (only if |r| < 1)

Example Problem & Step-by-Step Solution

Example 1 — Find an

Let a1=2, r=3, and n=5.

  1. Use an=a1 rn−1.
  2. Compute a5 = 2 · 34 = 2 · 81 = 162.

Example 2 — Find Sn

Let a1=10, r=1/2, and n=4.

  1. Use Sn=a1(1 − rn)/(1 − r).
  2. S4 = 10 · (1 − (1/2)4) / (1 − 1/2) = 10 · (1 − 1/16) / (1/2) = 10 · (15/16) / (1/2) = 18.75

Example 3 — Infinite sum

Let a1=6 and r=1/3.

  1. Since |r| = 1/3 < 1, the infinite series converges.
  2. S∞ = a1 / (1 − r) = 6 / (1 − 1/3) = 6 / (2/3) = 9

Frequently Asked Questions

Q: What makes a sequence geometric?

Each term is multiplied by the same constant ratio r to get the next term.

Q: When does an infinite geometric series converge?

It converges only when |r| < 1.

Q: What if r = 1?

Then every term equals a1, and the finite sum is Sn = n a1.

Q: Can I use fractions?

Yes. Turn on Prefer exact fractions to keep results exact when possible.