How is the nth term (a_n) calculated?

The nth term is given by a_n = a1 * r^(n-1), where a1 is the first term, r is the common ratio, and n is the position. For the sequence 3, 6, 12, ... the 8th term is 3 * 2^7 = 384.

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

The sum of the first n terms is S_n = a1 * (1 - r^n) / (1 - r) when r is not equal to 1. When r = 1 all terms are equal and S_n = a1 * n. For a decreasing series where |r| is less than 1, the infinite sum converges to S_inf = a1 / (1 - r).

When does an infinite geometric series converge?

An infinite geometric series converges (reaches a finite total) only when the absolute value of the common ratio is less than 1 (|r| < 1). If |r| is greater than or equal to 1 the partial sums grow without bound and the series diverges. For example, the series 1, 0.5, 0.25, 0.125, ... has r = 0.5 and its infinite sum is 1 / (1 - 0.5) = 2.

Can the common ratio be negative or a fraction?

Yes. A negative ratio produces an alternating sequence (terms flip sign each step), for instance 4, -8, 16, -32. A fractional ratio between 0 and 1 produces a decreasing sequence. The calculator handles all real non-zero values of r, including negative ratios where n - 1 is odd.

What is the difference between a geometric sequence and an arithmetic sequence?

In an arithmetic sequence consecutive terms differ by a constant amount (the common difference d), so each term is the previous term plus d. In a geometric sequence consecutive terms have a constant ratio (r), so each term is the previous term multiplied by r. Doubling repeatedly is geometric; adding 5 repeatedly is arithmetic.

Geometric Sequence Calculator

Q: What is a geometric sequence?

A geometric sequence (or geometric progression) is an ordered list of numbers in which each term after the first is found by multiplying the previous term by a fixed number called the common ratio r. For example, 3, 6, 12, 24, 48 is a geometric sequence with r = 2.

Q: Can the common ratio be negative or a fraction?

Yes. A negative ratio produces an alternating sequence (terms flip sign each step), for instance 4, -8, 16, -32. A fractional ratio between 0 and 1 produces a decreasing sequence. The calculator handles all real non-zero values of r, including negative ratios where n - 1 is odd.

Q: What is the difference between a geometric sequence and an arithmetic sequence?

In an arithmetic sequence consecutive terms differ by a constant amount (the common difference d), so each term is the previous term plus d. In a geometric sequence consecutive terms have a constant ratio (r), so each term is the previous term multiplied by r. Doubling repeatedly is geometric; adding 5 repeatedly is arithmetic.

A geometric sequence calculator that solves any of the four core unknowns in a geometric progression — the nth term, the first term, the common ratio, or the number of terms — plus the partial sum S_n and, when the series converges, the infinite sum S_inf. Every answer comes with full step-by-step working and a bar chart of the first terms so you can see the progression at a glance. Everything runs locally in your browser; nothing is sent to a server.

What is a geometric sequence?

A geometric sequence is a list of numbers where each term is obtained by multiplying the previous one by a fixed value called the common ratio (r). The general form is:

a1, a1·r, a1·r^2, a1·r^3, …, a1·r^(n-1)

The value of r determines the character of the sequence:

r > 1 — the terms grow exponentially (e.g., 2, 4, 8, 16, …)
0 < r < 1 — the terms shrink toward zero (e.g., 100, 50, 25, 12.5, …)
r < 0 — the terms alternate in sign (e.g., 3, -6, 12, -24, …)
r = 1 — all terms are identical (constant sequence)

How it works

Finding the nth term

The fundamental formula is:

a_n = a1 * r^(n-1)

Enter a1, r, and n; the calculator substitutes them and reports a_n together with the full expansion so you can check every step.

Finding the first term

Rearranging the formula:

a1 = a_n / r^(n-1)

Provide the nth term, the ratio, and the position n.

Finding the common ratio

If you know a1, a_n, and n:

r = (a_n / a1)^(1 / (n-1))

For a negative ratio the calculator checks whether n - 1 is odd before applying the negative root.

Finding the number of terms

Taking logarithms of both sides of a_n = a1 * r^(n-1):

n = log(a_n / a1) / log(r) + 1

The calculator verifies that the result is a positive integer; if your inputs are inconsistent it reports the exact decimal it obtained so you can spot the discrepancy.

Partial sum

The sum of the first n terms is:

S_n = a1 * (1 - r^n) / (1 - r) (for r not equal to 1) S_n = a1 * n (for r = 1)

Infinite sum

When |r| is less than 1 the partial sums converge and the infinite series has a finite value:

S_inf = a1 / (1 - r)

This is the limit as n tends to infinity. For |r| of 1 or more the series diverges and no finite total exists.

Worked example

Suppose you invest £500 and it grows by 8 % each year. The balance after n years is a geometric sequence with a1 = 500 and r = 1.08.

What is the balance after 10 years?

a_10 = 500 * 1.08^9 = 500 * 1.999005 = £999.50

What is the total of all 10 annual balances added together?

S_10 = 500 * (1 - 1.08^10) / (1 - 1.08) = 500 * (-1.158925) / (-0.08) = £7,243.28

a1	r	n	a_n	S_n
500	1.08	5	680.24	2,933.30
500	1.08	10	999.50	7,243.28
100	0.5	6	3.125	196.875
1	2	10	512	1,023

The infinite-sum formula is particularly useful for annuity and perpetuity valuation in finance, and for summing infinite series in mathematics.

Formula note

The derivation of S_n starts from the observation that r * S_n = a1 * r^n + a1 * r^(n+1) + … − first n-1 terms. Subtracting the original S_n from r * S_n cancels all interior terms and leaves S_n * (r - 1) = a1 * r^n - a1, which rearranges to the standard formula. The infinite-series result follows by letting n tend to infinity and noting that r^n tends to zero when |r| is less than 1.