Geometric Sequence Calculator

What Is a Geometric Sequence?

A geometric sequence is an ordered list in which every term after the first is produced by multiplying the previous term by the same constant. That constant is the common ratio, written as r. The sequence 3, 6, 12, 24, 48 is geometric because every term is twice the preceding term, so r = 2. The sequence 100, 50, 25, 12.5 is also geometric, but its ratio is 0.5 and its terms decrease toward zero.

This geometric sequence calculator uses the first term, common ratio, and term number to find the nth term and the sum of the first n terms. It also checks whether the corresponding infinite geometric series converges. A sequence describes the ordered terms; a series is what you get when those terms are added. Keeping that distinction clear prevents a common mistake when choosing a formula.

Geometric progressions model repeated proportional change. Compound growth, depreciation, population models, repeatedly scaled measurements, and idealized bouncing-ball distances can all follow geometric patterns. Real observations may only approximate a fixed ratio, so verify that a constant multiplier is reasonable before treating data as an exact geometric sequence.

How to Find the nth Term of a Geometric Sequence

The explicit nth-term formula is a_n = arⁿ⁻¹. Here a is the first term, r is the common ratio, and n is the position you want. The exponent is n − 1 because reaching the first term requires zero multiplications by r, reaching the second requires one multiplication, and reaching the nth requires n − 1 multiplications.

Suppose a = 2, r = 3, and n = 5. Substitute those values to get a₅ = 2 × 3⁴ = 2 × 81 = 162. A negative ratio uses the same formula. If a = 4 and r = −0.5, the signs alternate because odd powers of a negative number are negative and even powers are positive.

How to Find the Sum of a Finite Geometric Series

When r is not 1, the sum of the first n terms is S_n = a(1 − rⁿ) / (1 − r). An equivalent version is a(rⁿ − 1) / (r − 1); both produce the same answer. If r = 1, every term equals a, so use the direct rule S_n = na instead of dividing by zero in the general formula.

Using a = 2, r = 3, and n = 5 gives S₅ = 2(1 − 3⁵) / (1 − 3) = 2(−242)/(−2) = 242. This result is the total 2 + 6 + 18 + 54 + 162. The calculator selects the correct r = 1 branch automatically and retains full floating-point precision until the final display is formatted.

When Does an Infinite Geometric Series Converge?

An infinite geometric series has a finite sum only when |r| < 1. Under that condition, successive powers of r approach zero, and the sum approaches S_∞ = a/(1 − r). If r is 1, greater than 1, or at most −1, the terms do not shrink toward zero, so no finite infinite sum exists.

Consider a = 12 and r = 0.25. The infinite sum is 12/(1 − 0.25) = 16. For the first four terms, a₄ = 12 × 0.25³ = 0.1875 and S₄ = 15.9375, already close to the limit of 16. A negative ratio can also converge: 4, −2, 1, −0.5, ... has r = −0.5 and approaches the finite sum 4/(1.5) = 8/3 while alternating above and below that limit.

Worked Geometric Sequence Examples

Example 1: exponential growth

For a = 2, r = 3, and n = 5, the fifth term is 2 × 3⁴ = 162. The finite sum is 2(1 − 3⁵)/(1 − 3) = 242. Since |3| is greater than 1, the infinite series diverges even though every finite partial sum remains valid.

Example 2: decay toward a finite limit

For a = 12, r = 0.25, and n = 4, the fourth term is 0.1875 and the first four terms total 15.9375. Because |0.25| is less than 1, the infinite sum exists and equals 12/(1 − 0.25) = 16.

Example 3: alternating signs

For a = 4 and r = −0.5, the terms are 4, −2, 1, −0.5, 0.25, and so on. The signs alternate, but the magnitudes shrink, so the series converges to 4/(1 − (−0.5)) = 8/3. A negative ratio does not automatically mean divergence; its absolute value controls convergence.

Geometric Sequence Calculator FAQ

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

The formula is a_n = arⁿ⁻¹, where a is the first term, r is the common ratio, and n is the term position. The exponent n − 1 counts how many times the first term must be multiplied by r.

How do you find the common ratio of a geometric sequence?

Divide any nonzero term by the term immediately before it: r = a_n/a_n−1. Check more than one consecutive pair when possible; a true geometric sequence produces the same ratio each time.

What is the formula for the sum of a geometric series?

For r ≠ 1, the first n terms sum to S_n = a(1 − rⁿ)/(1 − r). If r = 1, every term is a, so the sum simplifies to S_n = na.

When does an infinite geometric series converge?

It converges only when the common ratio satisfies |r| < 1. In that interval the terms approach zero and the infinite sum is S_∞ = a/(1 − r); outside it there is no finite infinite sum.

Can a geometric sequence have a negative common ratio?

Yes. A negative ratio makes consecutive nonzero terms alternate in sign. The related infinite series still converges when the ratio lies strictly between −1 and 1.

What happens when the common ratio is 1 or 0?

When r = 1, every term equals the first term and the finite sum is na; the infinite series does not converge unless discussing the trivial all-zero series separately. When r = 0, the sequence is a, 0, 0, ... and every finite sum is a, while the infinite sum formula also returns a.

What is the difference between an arithmetic and geometric sequence?

An arithmetic sequence adds a fixed difference to create each term, while a geometric sequence multiplies by a fixed ratio. Arithmetic behavior is linear in the term number; geometric behavior follows powers and can model exponential growth or decay.