How this geometric sequence calculator works
A geometric sequence is a list of numbers in which each term is obtained by multiplying the previous term by the same fixed value. That constant multiplier is the common ratio, written r. Starting from the first term a₁, you multiply by r again and again: a₁, a₁·r, a₁·r², a₁·r³, and so on.
To find any term directly, use the nth term formula aₙ = a₁·r^(n−1) — you multiply the first term by the common ratio (n−1) times. To add up the first n terms, use the sum formula Sₙ = a₁(1−rⁿ)/(1−r) when r ≠ 1; if r = 1 every term equals a₁, so the sum is simply Sₙ = n·a₁. When the common ratio satisfies |r| < 1, the terms shrink toward zero and the series has a finite infinite sum S∞ = a₁/(1−r); if |r| ≥ 1 there is no finite infinite sum. This calculator computes all of these and lists the terms (capped at 100 for readability).
Reference — common ratio and term examples:
| a₁ | r | Sequence start | a₅ | S₅ | S∞ |
|---|---|---|---|---|---|
| 2 | 3 | 2, 6, 18, 54, 162 | 162 | 242 | — |
| 1 | 2 | 1, 2, 4, 8, 16 | 16 | 31 | — |
| 16 | ½ | 16, 8, 4, 2, 1 | 1 | 31 | 32 |
| 3 | −2 | 3, −6, 12, −24, 48 | 48 | 33 | — |
Frequently asked questions
- What is a geometric sequence?
- A geometric sequence is a list of numbers where each term is found by multiplying the previous term by the same fixed value. That fixed multiplier is the common ratio, r. For example 2, 6, 18, 54, 162 is geometric because each term is 3 times the one before it.
- What is the nth term formula for a geometric sequence?
- The nth term is aₙ = a₁·r^(n−1), where a₁ is the first term, r is the common ratio and n is the position. You multiply the first term by the common ratio (n−1) times. With a₁ = 2 and r = 3, the 5th term is 2·3^(5−1) = 2·81 = 162.
- How do I find the sum of a geometric sequence?
- The sum of the first n terms is Sₙ = a₁(1−rⁿ)/(1−r) when r ≠ 1. If r = 1 every term equals a₁, so Sₙ = n·a₁. For a₁ = 2, r = 3, n = 5, Sₙ = 2(1−3⁵)/(1−3) = 2(1−243)/(−2) = 242.
- What is the common ratio?
- The common ratio, r, is the constant value you multiply each term by to get the next term. You find it by dividing any term by the term before it. If |r| > 1 the terms grow, if |r| < 1 they shrink toward zero, and a negative r makes the signs alternate.
- When does an infinite geometric series converge?
- An infinite geometric series converges only when |r| < 1, that is when −1 < r < 1. Then the terms shrink toward zero fast enough that the total settles on a finite value S∞ = a₁/(1−r). If |r| ≥ 1 the terms do not shrink and the sum has no finite value.
- What is the difference between a geometric and an arithmetic sequence?
- In a geometric sequence you multiply by a fixed common ratio r, so terms grow or shrink multiplicatively (2, 6, 18, 54…). In an arithmetic sequence you add a fixed common difference d, so terms change by the same amount each step and grow linearly (2, 5, 8, 11…).