Geometric Sequence Calculator – nth Term & Sum

Geometric sequences grow by multiplication, and the calculator exposes both directions. For 3, 6, 12, 24, …: a₁ = 3 and r = 2, so the 10th term is a₁₀ = 3×2⁹ = 3×512 = 1536 and the ten-term sum is S₁₀ = 3(2¹⁰−1)/(2−1) = 3×1023 = 3069. Give it scattered terms instead — a₂ = 6 and a₅ = 48 — and it recovers r³ = 8, r = 2, a₁ = 3 and rebuilds the list. When |r| < 1 the infinite sum converges: for 8 + 4 + 2 + … the total is a₁/(1−r) = 8/0.5 = 16, the math behind repeating decimals like 0.999… = 1. The tool prints each power of r, warns when |r| ≥ 1 makes S∞ diverge, and switches to scientific notation when terms explode past a₂₀.

1. 1

   Enter a₁ and r

   Or provide two terms; the ratio is solved automatically

2. 2

   Pick the term index

   aₙ = a₁·rⁿ⁻¹ is evaluated with each power of r shown

3. 3

   Sum the first n terms

   Sₙ = a₁(rⁿ−1)/(r−1) — a special note appears for r = 1

4. 4

   Check convergence

   If |r| < 1, the infinite sum a₁/(1−r) is reported

• 1

  r = 1 is the degenerate case: every term equals a₁ and Sₙ = n·a₁

• 2

  |r| < 1 converges; |r| ≥ 1 infinite sums do not exist

• 3

  Compound interest is just a geometric sequence with r = 1 + rate

• Raising r to n instead of n−1, doubling the 10th term of 3,6,12,… to 3072

• Dividing by (1−r) when r > 1, where the finite formula needs (r−1)

• Calling 2, 4, 6, 8 geometric — constant difference means arithmetic

• Summing infinite series with r = −2, which oscillates and diverges