Geometric Sequence Calculator

What the Geometric Sequence Calculator Does

This tool finds the nth term and the sum of a geometric sequence from three inputs: the first term, the common ratio, and how many terms you want. A geometric sequence is a list of numbers where each term is found by multiplying the previous one by the same fixed number, called the common ratio (r).

It is useful for math and algebra students checking homework, anyone studying for exams, and people running quick calculations involving repeated multiplication, such as compound growth, depreciation, or doubling and halving patterns.

How It Works: The Geometric Sequence Formulas

The nth term of a geometric sequence uses the first term a1, the common ratio r, and the position n:

nth term: an = a1 * r^(n - 1)

Note the exponent is (n - 1), not n, because the first term is multiplied by r zero times. The sum of the first n terms (a finite geometric series) has its own formula:

• Sum when r is not 1: Sn = a1 * (1 - r^n) / (1 - r)
• Sum when r = 1: Sn = a1 * n (every term is identical)
• Infinite sum, only when |r| < 1: S = a1 / (1 - r)

Worked Example With Real Numbers

Suppose a1 = 3, r = 2, and you want the 5th term. Using an = a1 * r^(n - 1): a5 = 3 * 2^(5 - 1) = 3 * 2^4 = 3 * 16 = 48. The sequence is 3, 6, 12, 24, 48.

Now find the sum of those 5 terms. Using Sn = a1 * (1 - r^n) / (1 - r): S5 = 3 * (1 - 2^5) / (1 - 2) = 3 * (1 - 32) / (-1) = 3 * (-31) / (-1) = 93. Check by adding: 3 + 6 + 12 + 24 + 48 = 93. The results match.

Common Mistakes to Avoid

Most errors come from small slips in the inputs rather than the math itself. Watch for these:

• Using n instead of (n - 1) in the exponent. The first term has exponent 0.
• Confusing geometric with arithmetic sequences. Geometric multiplies by r; arithmetic adds a fixed difference d.
• Finding the ratio wrong: r = any term divided by the one before it (term2 / term1), not term1 / term2.
• Applying the infinite-sum formula when |r| >= 1. That series does not converge, so it has no finite sum.
• Forgetting the sign when r is negative, which makes terms alternate positive and negative.

Factors That Affect the Result

The common ratio drives the behavior of the whole sequence. When |r| > 1 the terms grow larger without bound; when |r| < 1 they shrink toward zero; and when r is negative the signs alternate. A ratio of exactly 1 produces a constant sequence, and a ratio of 0 collapses everything after the first term to zero.

Because terms grow or shrink so quickly, even a small change in r or n can move the answer dramatically, so double-check those two values first if a result looks off.