Geometric Sequence Calculator
Enter the first term and the common ratio, or two known terms. You get the nth term, the sum of n terms, the sum to infinity when it exists, and a list of terms.
Worked examples
Term and sum
First term 2, ratio 3, n = 10
Term 39,366, sum 59,048
2 × 3⁹ = 39,366, and 2 × (3¹⁰ - 1) ÷ 2 = 59,048.
Sum to infinity
First term 1, ratio 1/2
2
1 ÷ (1 - 1/2) = 2.
About Geometric Sequence Calculator
A geometric sequence multiplies by the same number each time, called the common ratio. The sequence 2, 6, 18, 54 has a ratio of 3. The nth term is the first term times the ratio to the power of n - 1, so the 10th term is 2 × 3⁹, which is 39,366. The sum of the first n terms is the first term times (1 - rⁿ) divided by (1 - r), which for ten terms of this sequence is 59,048.
If the ratio lies between -1 and 1, the terms shrink towards zero and the total of all of them, to infinity, settles on a fixed value: the first term divided by 1 - r. For 1, 1/2, 1/4 and so on, the sum to infinity is 2. If you know two terms rather than the first, the page finds the ratio by taking a root. When the gap between the positions is even, both a positive and a negative ratio can fit, and the page shows both. Whole and fractional values are handled exactly in the first-term option.
How to use Geometric Sequence Calculator
- Choose whether you know the first term and ratio, or two terms.
- Enter the values. Fractions such as 1/2 are fine.
- Enter n for the term and the sum you want.
- Read the nth term, the sum, the sum to infinity if it exists, and the list of first terms.
Common uses
- Finding the nth term in sequence homework
- Compound interest and growth patterns
- Bouncing ball, halving and doubling problems
- Working out a repeating decimal as a fraction
Frequently asked questions
What is the common ratio?
It is the number you multiply by to get from one term to the next. Divide any term by the one before it to find it.
When does a sum to infinity exist?
Only when the ratio is between -1 and 1, so that the terms get smaller and smaller. For other ratios the terms grow and the total has no limit.
Why can two ratios fit two known terms?
If the gap between the positions is even, a positive and a negative ratio give the same result when raised to that power. For example, 3 and -3 both square to 9, so both fit.
How is a geometric sequence linked to compound interest?
Each year the balance is multiplied by 1 plus the rate, so the balances form a geometric sequence with that as the ratio.
What if the ratio is negative?
The terms alternate between positive and negative, as in 2, -4, 8, -16.