Geometric Sequence Calculator
Enter the first term, the common ratio, and how many terms you want — get the nth term aₙ and the sum of the first n terms instantly.
Term and sum together
Enter a, r, and n and the calculator returns both the nth term aₙ = a·r^(n −
- and the running sum Sₙ of the first n terms.
n is a whole count
The number of terms n must be a whole number of at least 1 — there is no zeroth or fractional term in a sequence.
What is a geometric sequence?
A list with a constant ratio
A geometric sequence is a list of numbers in which each term is obtained by multiplying the one before it by a fixed factor — the common ratio r: a, ar, ar², … Because the ratio never changes, the whole sequence is pinned down by just two numbers, the first term a and the ratio r. This calculator turns those two values, plus a term count n, into the nth term aₙ and the sum Sₙ of the first n terms without you writing the list out by hand.
Enter the first term, the common ratio, and n to get the nth term and the sum of the first n terms.
The nth term comes from multiplying the first term by the ratio r raised to the power n − 1, and the sum follows from a compact closed-form formula.
aₙ = a × r^(n − 1) and Sₙ = a × (1 − rⁿ) ÷ (1 − r)First find the nth term by multiplying a by n − 1 copies of r. Then the sum follows from Sₙ = a(1 − rⁿ)/(1 − r) whenever r is not 1; when r equals 1 every term equals a, so the sum collapses to a·n. A ratio with |r| > 1 grows the terms, |r| < 1 shrinks them toward zero, and a negative r flips the sign at each step.
Suppose the sequence starts at a = 2 with a common ratio r = 3, and you want the 5th term and the sum of the first 5 terms.
Find the nth term
a₅ = 2 × 3^(5 − 1) = 2 × 81 = 162. The 5th term is 162.
Set up the sum
S₅ = 2 × (1 − 3⁵) ÷ (1 − 3) = 2 × (1 − 243) ÷ (−2) = 2 × (−242) ÷ (−2).
Compute the sum
S₅ = 2 × 121 = 242. The first five terms add up to 242.
The calculator hands back two numbers, and each answers a different question. The nth term aₙ tells you the single value sitting at position n — for 2, 6, 18, … the fifth term is 162, the number you would land on after four multiplications by three. The sum Sₙ tells you what all the terms from the first up to the nth add to: 242 for those same five terms. The common ratio r colours both outputs. When |r| is greater than 1 the terms and the running total grow ever faster, which is exactly the geometric growth behind compound interest and population models. When |r| is less than 1 the terms shrink toward zero and the sum closes in on a finite limit. A negative ratio alternates the sign of each term, so the sequence swings back and forth — and the sum can partly cancel out. Reading the size and sign of r tells you at a glance whether your sequence explodes, fades, or oscillates.
The formulas are exact, but a couple of practical points are worth keeping in mind.
n must be a whole number of at least 1
The number of terms n counts positions in the list, so it must be a whole number of 1 or more — the calculator returns no result for a fractional or zero n. The first term and the common ratio, by contrast, may be any numbers, positive or negative. This tool covers geometric sequences only, where each term is multiplied by a fixed ratio; sequences that add a constant amount each step are arithmetic and need a different formula.