Geometric Sequence Calculator

A geometric sequence is an ordered list in which every term after the first is found by multiplying the preceding term by a constant called the common ratio. If the first term is a and the ratio is r, the terms begin a, ar, ar squared, and so on. The nth term is a times r raised to the power n minus one. This calculator applies that definition directly and also evaluates finite and infinite geometric series. The common ratio controls the shape of the sequence. A ratio greater than one produces terms whose magnitudes grow. A positive ratio between zero and one produces terms that shrink toward zero. A negative ratio alternates the sign of consecutive terms, while a ratio of zero makes every term after the first equal to zero. When the ratio is exactly one, all terms equal the first term and the finite sum is simply the first term multiplied by the number of terms. A geometric series is the sum of terms in a geometric sequence. For a finite sequence, the sum of the first n terms is a times one minus r to the nth power, divided by one minus r. The special r equals one case is handled separately to avoid division by zero. Finite sums are valid for every real ratio and positive whole-number term count, so they are useful for repeated payments, depreciation schedules, branching patterns, and many algebra exercises. An infinite geometric series has a finite sum only when the absolute value of the common ratio is less than one. Under that condition, later terms become progressively smaller and their total approaches a divided by one minus r. If the ratio has magnitude one or greater, the partial sums do not settle at one finite value, so the calculator reports that the infinite series does not converge. This distinction is essential: a finite sum always stops after the requested number of terms, but an infinite sum describes a limiting process. Geometric sequences appear in compound interest, population models, radioactive decay, computer algorithms, music, and repeating decimals. Enter decimal or negative ratios when appropriate, and choose n as a positive integer. Results are rounded for display while calculations retain normal JavaScript floating-point precision. Comparing the nth term with the finite sum can help you see how individual terms contribute to the accumulated series.