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Geometric Sequence Calculator
Last updated: May 31, 2026
Written by Blake Boege
A geometric sequence calculator computes progression terms, finite sums, and infinite sums of geometric series. It uses formulas such as a_n = a_1 * r^(n-1) and S_n = a_1 * (1-r^n)/(1-r), checking for convergence (when |r| < 1) to solve the infinite sum. It displays a step-by-step analysis and the first ten terms of the sequence.
Find specific terms (a_n), finite sums (S_n), or infinite sums (S) of a geometric sequence. Includes step-by-step formula substitutions and a terms list table.
Quick Answer
Calculate the nth term, finite sum, or infinite sum of a geometric sequence. Enter the first term, common ratio, and term count to view the results.
Calculation Target
e.g. 2
e.g. 0.5
Must be a positive integer. · e.g. 10
Nth Term (a_10)
0.003906
a₁ = 2 · r = 0.5
A geometric sequence scales by a common ratio (r) at each step. Infinite geometric series only converge if |r| < 1.
Step-by-Step Formula Substitution
First 10 Terms of the Sequence
| Term (i) | Value (a_i) | Cumulative Sum (S_i) |
|---|---|---|
| 1 | 2 | 2 |
| 2 | 1 | 3 |
| 3 | 0.5 | 3.5 |
| 4 | 0.25 | 3.75 |
| 5 | 0.125 | 3.875 |
| 6 | 0.0625 | 3.9375 |
| 7 | 0.03125 | 3.96875 |
| 8 | 0.015625 | 3.984375 |
| 9 | 0.007813 | 3.992188 |
| 10 | 0.003906 | 3.996094 |
Examples
Find the 5th term for first term 2, common ratio 3
a_5 = 2 × 3^(5 - 1) = 2 × 81 = 162
Find the sum of the first 5 terms (a_1 = 2, r = 3)
S_5 = 2 × (1 - 3^5) / (1 - 3) = 242
Sum of infinite convergent series (a_1 = 4, r = 0.5)
S = 4 / (1 - 0.5) = 8 (converges because |0.5| < 1)
How it works
Geometric sequences scale by multiplying a constant ratio at each step:
Formula for the nth Term (a_n)
an = a1 · rn − 1
Formula for Finite Sum (S_n)
Sn = a1 · (1 − rn) / (1 − r)
Formula for Infinite Sum (S) — Only if |r| < 1
S = a1 / (1 − r)
Convergence vs. Divergence
One of the most interesting properties of geometric progressions is that a series with an infinite number of terms can sum up to a finite, exact number. This only occurs when the multiplier (common ratio) is fractional or decimal between -1 and 1. If the common ratio is greater than 1 or less than -1, the terms grow larger and larger, causing the infinite summation to diverge.
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Frequently asked questions
A geometric sequence is a list of numbers where each term after the first is found by multiplying the previous term by a constant value called the common ratio (r). For example, 2, 6, 18, 54... is a geometric sequence with a common ratio of 3.
The common ratio is the multiplier used to transition from one term to the next in a geometric sequence. It can be found by dividing any term in the sequence by the term that precedes it (r = a_n / a_(n-1)). The ratio can be positive, negative, fractional, or decimal.
An infinite geometric series converges (adds up to a finite number) if and only if the absolute value of the common ratio is strictly less than 1 (|r| < 1). If |r| ≥ 1, the terms do not shrink to zero fast enough, and the series diverges to infinity or oscillates indefinitely.
In an arithmetic sequence, terms change by adding a constant difference (e.g. +2 each step). In a geometric sequence, terms change by multiplying by a constant ratio (e.g. ×2 each step). Geometric sequences grow or shrink exponentially, whereas arithmetic sequences change linearly.
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