🔢 Geometric Sequence Calculator
By ToolNimba Math Team · Updated 2026-06-19
A geometric sequence is a list of numbers where each term is the one before it multiplied by a fixed number called the common ratio. This calculator takes your first term, the common ratio, and how many terms you want, then returns the nth term, the sum of the first n terms, the full list of terms, and the infinite sum when it exists. Type the three values and the answers update instantly, with every step you would write out by hand.
What is the Geometric Sequence Calculator?
A geometric sequence (also called a geometric progression) starts with a first term a and builds each later term by multiplying by a constant ratio r. So the terms are a, a·r, a·r^2, a·r^3, and so on. The thing that makes a sequence geometric is that the ratio between any two consecutive terms is always the same: divide any term by the one before it and you always get r. That is the test for whether a list of numbers is geometric at all.
Two formulas do most of the work. The nth term is a_n = a·r^(n-1), because to reach the nth term you multiply the first term by r a total of (n-1) times. The sum of the first n terms is S_n = a·(1 - r^n) / (1 - r) when r is not 1. When r equals 1 every term is just a, so the sum is simply a·n and the general formula is not used (it would divide by zero). These two results cover almost every textbook question about geometric series.
There is also an infinite sum. If the common ratio is between -1 and 1 (that is, the size of r is less than 1), the terms shrink toward zero fast enough that adding infinitely many of them still gives a finite answer: S∞ = a / (1 - r). If the size of r is 1 or larger, the terms do not shrink, the partial sums keep growing or oscillating, and the series diverges, so no finite infinite sum exists. This calculator reports the infinite sum only when it actually converges.
When to use it
- Checking homework on geometric progressions: finding a missing nth term, the common ratio, or a series sum.
- Modelling compound growth or decay, where a quantity is multiplied by the same factor each period.
- Working out the total of a repeating halving or doubling pattern, such as a bouncing ball or a folding problem.
- Confirming whether an infinite geometric series converges and, if so, what value it adds up to.
How to use the Geometric Sequence Calculator
- Enter the first term (a), the starting value of the sequence.
- Enter the common ratio (r), the fixed multiplier between consecutive terms.
- Enter the number of terms (n) you want to include.
- Read off the nth term, the sum of the first n terms, the term list, and the infinite sum if it converges.
Formula & method
Worked examples
First term a = 2, common ratio r = 3, find the 5th term and the sum of the first 5 terms.
- Write the terms: 2, 6, 18, 54, 162 (each is 3 times the one before).
- nth term = a x r^(n-1) = 2 x 3^(5-1) = 2 x 3^4 = 2 x 81 = 162
- Sum = a x (1 - r^n) / (1 - r) = 2 x (1 - 3^5) / (1 - 3)
- = 2 x (1 - 243) / (-2) = 2 x (-242) / (-2) = 242
- Check by adding: 2 + 6 + 18 + 54 + 162 = 242
Result: 5th term = 162, sum of first 5 terms = 242
First term a = 100, common ratio r = 0.5, find the 4th term, the sum of 4 terms, and the infinite sum.
- Write the terms: 100, 50, 25, 12.5 (each is half the one before).
- nth term = 100 x 0.5^(4-1) = 100 x 0.5^3 = 100 x 0.125 = 12.5
- Sum of 4 terms = 100 x (1 - 0.5^4) / (1 - 0.5) = 100 x (1 - 0.0625) / 0.5
- = 100 x 0.9375 / 0.5 = 187.5
- Since the size of r is less than 1, the infinite sum = a / (1 - r) = 100 / (1 - 0.5) = 100 / 0.5 = 200
Result: 4th term = 12.5, sum of 4 terms = 187.5, infinite sum = 200
Behaviour of a geometric sequence by common ratio r
| Common ratio r | What the terms do | Infinite sum |
|---|---|---|
| r greater than 1 | Grow without bound | Diverges (no finite sum) |
| r = 1 | Stay constant at a | Diverges (no finite sum) |
| 0 less than r less than 1 | Shrink toward zero, same sign | Converges to a / (1 - r) |
| -1 less than r less than 0 | Shrink toward zero, alternate sign | Converges to a / (1 - r) |
| r = -1 | Flip between a and -a | Diverges (no finite sum) |
| r less than -1 | Grow in size, alternate sign | Diverges (no finite sum) |
Example sequence with a = 3, r = 2
| Term number n | Formula a x r^(n-1) | Value | Running sum S_n |
|---|---|---|---|
| 1 | 3 x 2^0 | 3 | 3 |
| 2 | 3 x 2^1 | 6 | 9 |
| 3 | 3 x 2^2 | 12 | 21 |
| 4 | 3 x 2^3 | 24 | 45 |
| 5 | 3 x 2^4 | 48 | 93 |
Common mistakes to avoid
- Using n instead of n-1 in the nth term formula. The exponent is n-1, not n, because the first term has been multiplied by r zero times. With a = 2 and r = 3, the 5th term is 2 x 3^4 = 162, not 2 x 3^5 = 486.
- Confusing a geometric sequence with an arithmetic one. In a geometric sequence you multiply by a fixed ratio; in an arithmetic sequence you add a fixed difference. The pattern 2, 6, 18, 54 is geometric (times 3), while 2, 6, 10, 14 is arithmetic (plus 4).
- Expecting an infinite sum when r is too large. The infinite sum a / (1 - r) only exists when the size of r is strictly less than 1. If r is 1, -1, or bigger in size, the series diverges and has no finite total.
- Dividing by zero when r = 1. The sum formula a x (1 - r^n) / (1 - r) breaks when r = 1 because 1 - r = 0. When every term equals a, just use S_n = a x n instead.
Glossary
- Geometric sequence
- A list of numbers where each term is the previous term multiplied by a fixed common ratio.
- First term (a)
- The starting value of the sequence, the term you multiply by the ratio to build the rest.
- Common ratio (r)
- The fixed number that each term is multiplied by to get the next term.
- nth term
- The value at position n in the sequence, found with a_n = a x r^(n-1).
- Geometric series
- The sum of the terms of a geometric sequence, either a finite number of them or infinitely many.
- Convergence
- When an infinite series adds up to a finite value, which happens here only if the size of r is less than 1.
Frequently asked questions
What is a geometric sequence?
A geometric sequence is a list of numbers where each term equals the one before it multiplied by a fixed number called the common ratio. For example, 3, 6, 12, 24 is geometric with a common ratio of 2.
How do I find the nth term of a geometric sequence?
Use a_n = a x r^(n-1), where a is the first term, r is the common ratio, and n is the position. For a = 2 and r = 3, the 5th term is 2 x 3^4 = 162. This calculator computes it for you.
How do I find the common ratio?
Divide any term by the term immediately before it. If the answer is the same for every neighbouring pair, that value is the common ratio r. For 5, 15, 45 the ratio is 15 / 5 = 3.
What is the formula for the sum of a geometric series?
For the first n terms, S_n = a x (1 - r^n) / (1 - r) when r is not 1. When r equals 1, every term is a, so the sum is simply a x n.
When does an infinite geometric series have a sum?
Only when the size of the common ratio is less than 1, that is r is strictly between -1 and 1. In that case S = a / (1 - r). If r is 1 or larger in size, the series diverges and has no finite sum.
What is the difference between a geometric and an arithmetic sequence?
A geometric sequence multiplies by a fixed ratio each step, while an arithmetic sequence adds a fixed difference. So 2, 6, 18 is geometric (times 3) but 2, 6, 10 is arithmetic (plus 4).