Geometric Sequence Calculator
Written by Thierno Sadou Diallo, formula verified per our methodology • Last checked on 9/5/2026
The nth term of a geometric sequence is calculated as uₙ = u₁ × r^(n−1), and the sum of the first n terms as Sₙ = u₁ × (r^n − 1) ÷ (r − 1). For u₁=3, r=2, and n=5, the 5th term is 48 and the sum of the first 5 terms is 93.
Explanation
A geometric sequence is a sequence of numbers where each term is obtained by multiplying the previous one by a fixed value, called the common ratio (r) — unlike our arithmetic sequence calculator, where each term is obtained by adding a fixed value to the previous one. This difference between addition and multiplication fundamentally changes the sequence's behavior: a geometric sequence with a ratio greater than 1 grows exponentially (each term grows proportionally faster than the previous one), while a ratio between 0 and 1 instead makes the sequence decrease toward zero without ever reaching it. The sum of the first n terms follows a convenient closed-form formula, which avoids having to add term by term: it relies on the fact that multiplying the whole sum by the ratio and then subtracting it from itself makes all the intermediate terms disappear, leaving only the first and the last. The special case r = 1 (a constant sequence, where each term is identical to the previous one) is handled separately, since the general formula would then divide by zero: the sum in that case simply reduces to n times the value of the first term. Geometric sequences model many real phenomena with proportional growth or decay: compound interest (see our compound interest calculator, where the principal follows exactly a geometric progression with ratio (1+rate)), radioactive decay, or serial dilution in chemistry.
Example: u₁=3, r=2, n=5
Inputs
First term: 3. Common ratio: 2. Position: 5.
Calculation
u₅ = 3 × 2⁴ = 3 × 16 = 48. S₅ = 3 × (2⁵ − 1) ÷ (2 − 1) = 3 × 31 ÷ 1 = 93 (that is, 3+6+12+24+48 = 93).
Result
The 5th term is 48, and the sum of the first 5 terms is 93.
Frequently asked questions
What happens if the common ratio is negative?
A negative common ratio makes the sign of successive terms alternate (positive, negative, positive...), while still following the same formula uₙ = u₁ × r^(n−1). For example, with u₁=1 and r=−2, the sequence gives 1, −2, 4, −8, 16: the absolute value always grows geometrically, but the sign alternates with each term.
Why does the sum diverge if |r| is greater than 1 and n grows?
When |r| > 1, each term is larger in absolute value than the previous one, so the sum of the terms grows without bound as n increases — it has no finite value even when summing infinitely many terms. Conversely, when |r| < 1, each term becomes smaller and smaller, and the sum of all the terms (even infinitely many) converges to a precise finite value, equal to u₁ ÷ (1 − r).
How do I recognize that a sequence of numbers is geometric rather than arithmetic?
In a geometric sequence, the ratio between two consecutive terms (next term divided by previous term) stays constant from one term to the next — that's the common ratio r. In an arithmetic sequence, it's instead the difference between two consecutive terms that stays constant. Dividing several pairs of consecutive terms by each other lets you quickly check which of the two models applies to a given series of numbers.