Geometric Sequence Calculator
The geometric sequence calculator multiplies by the common ratio r. For 3, 6, 12, ... with r = 2, the 10th term is 3 x 2^9 = 1,536 and the first 10 terms sum to 3,069.
Term 10
- Term 10 a_n = a1 x r^(n - 1) = 3 x 2^9 = 1,536
- 1,536
- Sum of the first 10 terms S_n = a1 (1 - r^n) / (1 - r) = 3 (1 - 2^10) / (1 - 2) = 3,069
- 3,069
- Explicit formula Any term from its position, without listing the ones before it.
- a_n = 3 x 2^(n - 1)
- Recursive formula Each term from the one before it.
- a1 = 3, a_n = 2 x a_(n - 1)
- Sum to infinity Only a geometric series with -1 < r < 1 has a finite sum to infinity. With this ratio the terms do not shrink to zero, so the partial sums never settle.
- does not converge
- First 10 terms
- 3, 6, 12, 24, 48, 96, 192, 384, 768, 1536
| n | a_n | Sum to n |
|---|---|---|
| 1 | 3 | 3 |
| 2 | 6 | 9 |
| 3 | 12 | 21 |
| 4 | 24 | 45 |
| 5 | 48 | 93 |
| 6 | 96 | 189 |
| 7 | 192 | 381 |
| 8 | 384 | 765 |
| 9 | 768 | 1,533 |
| 10 | 1,536 | 3,069 |
- Every value is exact: fractions are kept as fractions and whole numbers of any length are computed in full. Past 40 digits a value is shown in scientific notation with its length.
How the Arithmetic Sequence Calculator works
A geometric sequence multiplies by the same number each time. In 3, 6, 12, 24, ... each term is twice the one before, so the common ratio is r = 2. The ratio can be a fraction (64, 32, 16, ... has r = 1/2) or negative (1, -3, 9, -27, ... has r = -3, and the signs alternate).
The nth term. To reach term n you multiply by r a total of n - 1 times:
a_n = a1 x r^(n - 1)
For a1 = 3 and r = 2, the 10th term is 3 x 2^9 = 3 x 512 = 1,536. The recursive formula is the same rule one step at a time: each term is r times the one before.
The sum of n terms. Write S = a1 + a1 r + ... + a1 r^(n - 1), multiply both sides by r, and subtract. Everything cancels except the first and last pieces, which leaves
S_n = a1 (1 - r^n) / (1 - r)
For the first 10 terms of 3, 6, 12, ... that is 3 x (1 - 1,024) / (1 - 2) = 3 x 1,023 = 3,069. That sum is a geometric series, so this geometric sequence calculator is a geometric series calculator too. When r = 1 the formula divides by zero, and the sum is simply n x a1.
The sum to infinity. If r is between -1 and 1, r^n shrinks toward zero as n grows, and the sum approaches a1 / (1 - r). For 64, 32, 16, ... that is 64 / (1 - 1/2) = 128: the terms never stop, but their total never passes 128. With r = 2 the terms grow without limit and there is no finite sum.
The common ratio from two terms. From term i to term j the value is multiplied by r exactly j - i times, so r^(j - i) is the second value divided by the first. If term 2 is 6 and term 5 is 48, r^3 = 8 and r = 2. When the gap is even there are two real answers, r and -r, because an even power hides the sign; when the quotient is negative and the gap even, there is no real ratio at all. The calculator finds the ratio exactly whenever it is a fraction and says so when it is irrational.
Exact values. Geometric sequences grow fast and shrink fast. With r = 1/3 the 20th term is 2/1,162,261,467, and the calculator keeps it as that fraction rather than a decimal with most of its digits lost.
Arithmetic Sequence Calculator FAQ
What is the formula for the nth term of a geometric sequence?
a_n = a1 x r^(n - 1), where a1 is the first term and r the common ratio. For 3, 6, 12, ... the 10th term is 3 x 2^9 = 1,536.
How do you find the sum of a geometric sequence?
S_n = a1 (1 - r^n) / (1 - r) for r not equal to 1. The first 10 terms of 3, 6, 12, ... sum to 3 x (1 - 1,024) / (1 - 2) = 3,069.
When does a geometric series have a sum to infinity?
When the common ratio is strictly between -1 and 1. The sum is then a1 / (1 - r): 64 + 32 + 16 + ... = 64 / (1 - 1/2) = 128.
How do you find the common ratio?
Divide any term by the one before it. From two terms that are not next to each other, divide the later by the earlier and take the root matching the gap in positions: if term 2 is 6 and term 5 is 48, r^3 = 8, so r = 2.