Geometric Sequence Calculator
Calculate geometric sequence instantly with our math tool. Shows detailed work, formulas used, and multiple solution methods.
Reviewed by Manoj Kumar, Mathematics Educator
Formula
a_n = a * r^(n-1) | S_n = a(1 - r^n)/(1 - r)
Where a is the first term, r is the common ratio, n is the term position. For infinite series with |r| < 1, the sum converges to S = a/(1-r).
Worked Examples
Example 1: Finding the 10th Term and Sum
Problem:Find the 10th term and sum of the first 10 terms of the geometric sequence with a = 3 and r = 2.
Solution:a_10 = 3 * 2^(10-1) = 3 * 2^9 = 3 * 512 = 1,536\nS_10 = 3 * (1 - 2^10) / (1 - 2)\nS_10 = 3 * (1 - 1024) / (-1)\nS_10 = 3 * 1023 = 3,069\nSequence: 3, 6, 12, 24, 48, 96, 192, 384, 768, 1536
Result:10th term = 1,536 | Sum of 10 terms = 3,069
Example 2: Convergent Infinite Series
Problem:Find the sum of the infinite geometric series: 100 + 50 + 25 + 12.5 + ...
Solution:First term a = 100, common ratio r = 50/100 = 0.5\nSince |r| = 0.5 < 1, the series converges.\nS = a / (1 - r) = 100 / (1 - 0.5)\nS = 100 / 0.5 = 200\nVerification: S_10 = 100 * (1 - 0.5^10) / 0.5 = 199.8 (very close to 200)
Result:Infinite sum = 200 | The series converges since |r| = 0.5 < 1
Frequently Asked Questions
What is a geometric sequence and how is it defined?
A geometric sequence (also called a geometric progression) is an ordered list of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio r. The general form is a, ar, ar^2, ar^3, and so on. For example, the sequence 3, 6, 12, 24, 48 is geometric with first term a = 3 and common ratio r = 2. Each term is exactly twice the previous one. Geometric sequences can grow (when |r| > 1), shrink toward zero (when |r| < 1), alternate in sign (when r < 0), or remain constant (when r = 1). They appear naturally in finance, biology, physics, and computer science.
How do you find the nth term of a geometric sequence?
The nth term of a geometric sequence is given by the formula a_n = a * r^(n-1), where a is the first term, r is the common ratio, and n is the position number. For example, in the sequence with a = 5 and r = 2, the 8th term is 5 * 2^7 = 5 * 128 = 640. This formula works because reaching the nth term requires multiplying by r exactly (n-1) times starting from the first term. When calculating large term numbers, the result can grow extremely fast for |r| > 1 or shrink rapidly for |r| < 1. This exponential nature is what distinguishes geometric sequences from arithmetic sequences, where each term differs by a constant addition.
What is the formula for the sum of a geometric series?
The sum of the first n terms of a geometric series is S_n = a * (1 - r^n) / (1 - r) when r is not equal to 1, and S_n = a * n when r = 1. For an infinite geometric series with |r| < 1, the sum converges to S = a / (1 - r). For example, the infinite series 1 + 1/2 + 1/4 + 1/8 + ... has a = 1 and r = 1/2, giving S = 1/(1 - 0.5) = 2. The finite sum formula is derived by multiplying S_n by r and subtracting from S_n to eliminate most terms. The infinite sum exists only when |r| < 1 because the terms diminish to zero fast enough that the total remains bounded.
What determines whether a geometric series converges or diverges?
A geometric series converges if and only if the absolute value of the common ratio is less than 1 (|r| < 1). When |r| < 1, each successive term becomes smaller and smaller, approaching zero, and the partial sums approach a finite limit of a/(1-r). When |r| >= 1, the terms do not approach zero, so the series diverges (the partial sums grow without bound or oscillate). When r = -1, the series oscillates between two values and does not converge. When |r| = 1 and r is not -1, the series grows linearly. This convergence criterion is one of the simplest and most important tests in series analysis and provides a foundation for understanding more complex convergence tests.
References
Reviewed by Manoj Kumar, Mathematics Educator · Editorial policy