Geometric Sequence Calculator
A geometric sequence multiplies by the same fixed ratio at every step rather than adding a fixed amount — the pattern behind compound interest, viral growth, and radioactive decay, depending on whether the ratio is above or below 1.
Inputs
Result
aₙ = 4,374
Sum = 6,560
How the geometric sequence calculator works
The nth term: aₙ = a × r^(n−1), where a is the first term and r is the common ratio.
The sum of the first n terms (r ≠ 1): Sₙ = a × (r^n − 1) ÷ (r − 1). When r = 1, every term is identical, so the sum is simply a × n.
Worked example: first term 2, common ratio 3, 8th term
- a₈ = 2 × 3^7 = 2 × 2,187 = 4,374.
- Sum of first 8 terms = 2 × (3^8 − 1) ÷ (3 − 1) = 2 × 6,560 ÷ 2 = 6,560.
Common mistakes to avoid
Using (n) instead of (n−1) in the exponent
The first term a₁ has zero multiplications by r: a × r^(1−1) = a × 1 = a. Forgetting the −1 in the exponent throws off every term by one extra factor of r.
Forgetting the special case r = 1
When the ratio is exactly 1, every term is identical and the standard sum formula divides by zero — the sum is simply a × n instead.
Frequently asked questions
What does a common ratio between 0 and 1 mean for the sequence?
The terms shrink toward zero rather than grow — this is the pattern behind exponential decay, like a bouncing ball losing height with each bounce.
How is this related to compound interest?
Compound interest is exactly a geometric sequence: the balance each year is the previous balance multiplied by (1 + rate), making the rate-plus-one the common ratio.
Can the common ratio be negative?
Yes — this makes the sequence alternate in sign every term (positive, negative, positive...) while still growing or shrinking in magnitude according to |r|.
What happens to the sum as n grows very large, if r is between -1 and 1?
The sum converges to a finite limit, a ÷ (1 − r), rather than growing without bound — this is the basis of the 'infinite geometric series' result.
Related calculators
Arithmetic Sequence Calculator
nth term and sum of an arithmetic progression.
Compound Interest Calculator
See how money compounds over time with any compounding frequency.
Exponent Calculator
Raise any base to any power.
Percent Change Calculator
Increase or decrease between two numbers.
Rule of 72 Calculator
How many years until your money doubles.