A geometric series calculator finds the nth term and the sum of a geometric series. Enter the first term, the common ratio and the number of terms. It returns the nth term, the sum of the first n terms, the first few terms of the sequence and, when the ratio allows it, the sum of the infinite series.
In a geometric sequence each term is the previous term multiplied by the same number, called the common ratio. The sequence 3, 6, 12, 24 has a common ratio of 2.
This calculator needs JavaScript. Please enable it in your browser.
How the geometric series calculator works
The formulas
a_n = a1 × r^(n − 1)
S_n = a1 × (1 − r^n) ÷ (1 − r)
Where:
- a1 = the first term
- r = the common ratio
- n = the number of terms
- a_n = the nth term
- S_n = the sum of the first n terms
When the ratio is 1
If r = 1, every term equals a1, so the sum is simply n × a1. The formula above would divide by zero in that case.
Infinite geometric series
If the ratio is between −1 and 1, the terms shrink toward 0 and the infinite series has a finite sum.
S = a1 ÷ (1 − r) when |r| < 1
If |r| is 1 or more, the terms do not shrink and the infinite series has no finite sum.
Finding the common ratio
Divide any term by the term before it. In 3, 6, 12, 24, the ratio is 6 ÷ 3 = 2 and also 12 ÷ 6 = 2. If the ratios are not all the same, the sequence is not geometric.
Worked examples
Example 1: a1 = 3, r = 2, n = 8
The terms are 3, 6, 12, 24, 48, 96, 192, 384.
a_8 = 3 × 2^7 = 3 × 128 = 384
S_8 = 3 × (1 − 2^8) ÷ (1 − 2) = 3 × (−255) ÷ (−1) = 765
Adding the eight terms gives the same 765, which checks the formula. Because r = 2 is greater than 1, the infinite series has no finite sum.
Example 2: a1 = 81, r = 0.5, n = 6
The terms are 81, 40.5, 20.25, 10.125, 5.0625, 2.53125.
a_6 = 81 × 0.5^5 = 2.53125
S_6 = 81 × (1 − 0.5^6) ÷ (1 − 0.5) = 159.46875
Since |r| < 1, the infinite sum exists: S = 81 ÷ (1 − 0.5) = 162. The first six terms already add up to 159.47 of that 162.
Example 3: A ratio of 1
With a1 = 5, r = 1 and n = 4, every term is 5, so S_4 = 4 × 5 = 20.
Frequently asked questions
What is a geometric series?
It is the sum of the terms of a geometric sequence, where each term is the previous one multiplied by a fixed common ratio.
What is the difference between a sequence and a series?
A sequence is the list of terms, such as 3, 6, 12, 24. A series is their sum, such as 3 + 6 + 12 + 24.
When does an infinite geometric series have a sum?
Only when the absolute value of the common ratio is less than 1. The sum is a1 ÷ (1 − r).
How do I find the nth term?
Multiply the first term by the ratio raised to the power n − 1. For the eighth term of 3, 6, 12 and so on, that is 3 × 2^7 = 384.
What if the common ratio is negative?
The terms alternate in sign, and the same formulas apply. For example, with a1 = 2 and r = −1 the terms are 2, −2, 2, −2.