Geometric Sequence Calculator
List every term, find the Nth term and sum a geometric sequence from its first term and ratio or any two known terms, including fractional or negative ratios.
Fractions such as 5/2 are fine.
2, 6, 18, 54
54
80
What this geometric sequence calculator does
Enter the first term, the common ratio, and how many terms you want, and this tool instantly lists the whole sequence, highlights the Nth (last) term, and adds everything up into a total. A geometric sequence is any list of numbers where each term is found by multiplying the one before it by the same fixed number — the common ratio — such as 2, 6, 18, 54, where the ratio is 3. The ratio can be greater than one for a growing sequence, between zero and one for a shrinking one, or negative to make the signs alternate, and both the first term and the ratio may be decimals or fractions such as 1/2. If you know two terms instead, set "Start from" to "Two known terms" and it finds the ratio and the first term from them. If each term is instead the one before plus a fixed difference, use the Arithmetic Sequence Calculator.
The formulas behind the results
The Nth term is computed directly with a₁ × r^(N − 1), so you get the value of the last term without stepping through every multiplication. The sum uses the closed-form series formula a₁ × (r^N − 1) / (r − 1); when the ratio is exactly 1 every term is the same, so the sum is simply a₁ × N, which the tool handles as a special case to avoid dividing by zero. Everything recalculates live as you type, and each result has its own copy button. A "Working" block shows each formula with your own numbers in it, and a chart of the terms says whether the sequence is growing, decaying towards zero, oscillating or constant. "Decimal places" rounds every term and answer to the places you pick, and "List from term" lists just a stretch, a_k to a_N, with a "Sum of terms k to N" answer beside the others. “Reset the form” puts every field back where the page opened. To keep the listing readable and responsive, the number of terms is limited to 1000.
Frequently asked questions
How is the Nth term different from the sum?
The Nth term is a single value, a₁ × r^(N − 1). The sum is every term up to it added together, a₁ × (r^N − 1) ÷ (r − 1); at a ratio of exactly 1 that would divide by zero, so the page uses a₁ × N.
Can the common ratio be negative, a fraction, or exactly one?
Yes. A negative ratio makes the signs alternate, a ratio between 0 and 1 makes the sequence shrink toward zero, and a ratio of exactly 1 repeats the first term so the sum becomes first term × N.
What is the 'sum to infinity'?
When the common ratio is between −1 and 1 (|r| < 1) the terms shrink toward zero and the series converges to a finite total even with infinitely many terms: a₁ / (1 − r). The tool shows this 'Sum to ∞' row only in that case — for a ratio of 1/2 starting at 1, it equals 2. When |r| ≥ 1 the series diverges, so the row is hidden.
Can I find the ratio from two terms I know?
Yes. Set "Start from" to "Two known terms" and fill in "Position p", "Term at p", "Position q" and "Term at q": the ratio is the (q − p)-th root of a_q ÷ a_p, the first term follows from it, and the "Working" block prints both lines with your numbers. When the two terms are an even number of places apart, the negative ratio fits as well and the page says so; a known term of 0, or an even gap between terms of opposite sign, has no real ratio and is refused with the reason.
Can I list only part of the sequence, or round the answers?
Yes. "List from term" lists a stretch from a_k to a_N and adds a "Sum of terms k to N" answer, worked from the closed form started at a_k; "Decimal places" rounds every term and answer to the places you pick, or leaves them as entered.