Geometric Sequence Calculator
The geometric sequence calculator finds the nᵗʰ term and the sum of a geometric sequence (to infinity if possible).
Term 1039,366
Sum of the first 1059,048
Common ratio3
Sum to infinitydivergesneeds |r| < 1 to converge
Behaviourgrows without bound
Terms
| n | Term |
|---|---|
| 1 | 2 |
| 2 | 6 |
| 3 | 18 |
| 4 | 54 |
| 5 | 162 |
| 6 | 486 |
| 7 | 1,458 |
| 8 | 4,374 |
| 9 | 13,122 |
| 10 | 39,366 |
The formula
aₙ = a₁rⁿ⁻¹, Sₙ = a₁(1 − rⁿ) ÷ (1 − r)
Convergence depends only on r
An infinite geometric series converges exactly when |r| < 1, and then sums to a₁ ÷ (1 − r). That is why 1 + ½ + ¼ + ⅛ + … equals exactly 2, and why 0.999… equals exactly 1 — it is the geometric series with a₁ = 0.9 and r = 0.1.