Geometric Sequence Calculator

A geometric sequence is a 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 foundational formula for finding the nth term (an) of a geometric sequence is an = a1 * r^(n - 1), where a1 represents the first term and n is the position of the term. To find the sum of a finite geometric sequence (Sn), the calculator uses the formula Sn = a1 * (1 - r^n) / (1 - r) when r != 1. For infinite sums where the absolute value of the common ratio is less than 1 (|r| < 1), the sum converges to S = a1 / (1 - r).

Let us walk through a practical example. Suppose you have a geometric sequence where the first term (a1) is 3, and the common ratio (r) is 2. You want to find the 5th term (n = 5) and the sum of the first 5 terms. First, apply the nth term formula: a5 = 3 * (2)^(5 - 1) = 3 * (2)^4 = 3 * 16 = 48. Next, calculate the finite sum using the sum formula: S5 = 3 * (1 - 2^5) / (1 - 2) = 3 * (1 - 32) / (-1) = 3 * (-31) / (-1) = 93. Thus, the 5th term is 48 and the sum of the first 5 terms is 93.

Always verify whether your common ratio is a fraction or a whole number before computing powers, as negative or fractional ratios lead to alternating signs or rapid decay. When dealing with two non-consecutive terms, you can find the common ratio by taking the nth-m root of the quotient of the two terms. Be careful with infinite sums: if your common ratio is greater than or equal to 1 in magnitude, the series diverges, and the sum is undefined.