A geometric sequence is a list of numbers where each term is found by multiplying the previous term by the same fixed value, called the common ratio r. If the first term is a and the common ratio is r, the sequence is a, ar, ar squared, ar cubed, and so on. The nth term is a times r to the power of (n minus 1), allowing you to jump directly to any position in the sequence without computing all the ones before it. The partial sum of the first n terms has its own formula: S(n) = a times (1 minus r to the n) divided by (1 minus r), which applies whenever r is not equal to 1. When r equals 1, every term is the same and the sum is simply n times a. Geometric sequences model many real-world situations: compound interest growth, population growth at a constant rate, radioactive decay, the bouncing height of a ball that loses a fixed percentage of its height with each bounce, and the depreciation of an asset by a fixed percentage each year. When the absolute value of the ratio r is less than 1, the terms shrink toward zero and the infinite series converges to a finite limit of a divided by (1 minus r). You enter the first term a, the common ratio r, and the number of terms n, and the calculator returns the nth term, the partial sum, the infinite sum when the series converges, and a full list of all n terms. The first term can be any non-zero decimal and r can be any signed decimal; n must be a positive integer. The terms list is shown for n up to 50.
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162
nth term a(n)
Partial sum S(n)242
Infinite sum S(∞)Diverges
Last term162
2, 6, 18, 54, 162
Infinite sum only exists when |r| < 1. Formula: a(n) = a × r^(n-1). Sum: S(n) = a(1 - r^n)/(1 - r).
How it works
The nth term is a(n) = a times r to the power of (n minus 1). The partial sum is S(n) = a times (1 minus r to the n) divided by (1 minus r) when r is not equal to 1; when r equals 1 the sum is n times a. The infinite sum is a divided by (1 minus r) when the absolute value of r is strictly less than 1; when the absolute value of r is 1 or more the series diverges and no finite infinite sum exists. The terms are generated by repeatedly multiplying by r, starting from a.
Worked example
Using the defaults a=2, r=3, n=5: the terms are 2, 6, 18, 54, 162. The nth term is 2 times 3 to the power of 4 = 2 times 81 = 162. The partial sum is 2 times (1 minus 3 to the 5) divided by (1 minus 3) = 2 times (1 minus 243) divided by (-2) = 2 times (-242) divided by (-2) = 242. Because |r| = 3 is greater than 1, the series diverges and there is no infinite sum.
