Geometric Sequence Calculator — nth term and sum

How it works

A geometric sequence grows by repeated multiplication: each term is the one before it times a constant common ratio r. Starting from the first term a₁, the terms are a₁, a₁r, a₁r², a₁r³, … — so the exponent on r is always one less than the term's position:

aₙ = a₁ · r^(n − 1)

To add up the first n terms you do not have to compute them one by one. The geometric series has a closed form:

Sₙ = a₁ · (1 − rⁿ) ÷ (1 − r) (for r ≠ 1)

When r = 1 every term equals a₁, so the sum is just a₁ · n. A ratio between −1 and 1 shrinks the terms toward zero; a ratio with magnitude greater than 1 makes them grow without bound; and a negative ratio makes the signs alternate.

This differs from an arithmetic sequence, which adds a constant difference instead of multiplying by a constant ratio.

Worked example

Take a₁ = 2 and r = 3, giving the sequence 2, 6, 18, 54, 162, … The 5th term is 2 · 3^(5−1) = 2 · 3⁴ = 2 · 81 = 162. The sum of the first five terms is 2 · (1 − 3⁵) ÷ (1 − 3) = 2 · (1 − 243) ÷ (−2) = 2 · (−242) ÷ (−2) = 242, which you can check by adding 2 + 6 + 18 + 54 + 162 = 242.

Frequently asked questions

What is the formula for the nth term of a geometric sequence?

The nth term is aₙ = a₁ · r^(n − 1), where a₁ is the first term and r is the common ratio. The exponent is n − 1 because the first term (n = 1) is multiplied by r zero times. For a₁ = 2 and r = 3, the 4th term is 2 · 3³ = 54.

How do I find the sum of a geometric sequence?

Use Sₙ = a₁ · (1 − rⁿ) ÷ (1 − r) for the first n terms, as long as r ≠ 1. If r = 1 the terms are all equal, so the sum is a₁ · n. The formula avoids adding the terms one at a time, which matters when n is large.

What is the common ratio?

The common ratio r is the number you multiply by to move from one term to the next, so you can find it by dividing any term by the term before it: r = aₙ ÷ aₙ₋₁. In 3, 6, 12, 24 the ratio is 6 ÷ 3 = 2.

How is a geometric sequence different from an arithmetic one?

A geometric sequence **multiplies** by a constant ratio, so it grows or shrinks exponentially; an arithmetic sequence **adds** a constant difference, so it changes by the same amount each step. 2, 6, 18, 54 is geometric (× 3); 2, 5, 8, 11 is arithmetic (+ 3).

Does the sum converge if the ratio is a fraction?

For a **finite** number of terms the sum is always a₁(1 − rⁿ) ÷ (1 − r). If |r| < 1 and you keep going forever, rⁿ approaches 0 and the infinite sum approaches a₁ ÷ (1 − r) — a finite limit. This calculator reports the finite sum of the first n terms.

Can the ratio or first term be negative?

Yes. A negative ratio makes the signs alternate — a₁ = 3, r = −2 gives 3, −6, 12, −24 — and a negative first term flips the whole sequence. The same formulas apply throughout.