Geometric Sequence Calculator
The nth term and the sum of a geometric sequence.
Result
nth term
162.00
Sum of n terms
242.00
How it works
A geometric sequence multiplies by the same factor at each step: start at a₁ and multiply by the common ratio r every time. The nth term is a₁ · r^(n−1), and the sum of n terms is a₁(rⁿ−1)/(r−1). Multiplying rather than adding is what makes these sequences explode. Starting at 1 with r = 2 gives 1, 2, 4, 8, 16 — the tenth term is already 512 and the twentieth 524,288. This is the mathematics behind compound interest, viral spread, population growth and the famous chessboard-and-grains story. When |r| < 1 each term shrinks, and something remarkable happens: an infinite number of terms adds up to a finite total, a₁/(1−r). Halving forever — 1 + ½ + ¼ + … — converges to exactly 2. That convergence underpins everything from repeating decimals to the maths of perpetuities.
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Frequently asked questions
What is the common ratio?
The fixed factor multiplied between consecutive terms — for example 3 in 2, 6, 18, 54.
When does the sum converge?
For an infinite series it converges only when the ratio is between −1 and 1.
What is the difference from an arithmetic sequence?
An arithmetic sequence adds a fixed amount each step (2, 5, 8, 11), a geometric one multiplies by a fixed factor (2, 6, 18, 54). Adding grows in a straight line; multiplying grows exponentially.
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