Geometric Sequence Calculator
Geometric Sequence Calculator
A geometric sequence multiplies by the same fixed ratio to get from one term to the next, which means it can grow (or shrink) far more dramatically than an arithmetic sequence over the same number of steps. Enter the first term, the common ratio, and which term number you want, and this calculator finds that term directly along with the sum of all terms up to that point. It's especially useful for modeling compounding processes, where each step scales by a constant factor rather than adding a constant amount.
- In a geometric sequence, each term is found by multiplying the previous term by a constant common ratio, e.g. 2, 6, 18, 54, 162… has a common ratio of 3.
- The nth term formula skips ahead directly: aₙ = a₁ × r^(n−1), avoiding the need to multiply repeatedly.
- The sum of the first n terms uses the formula Sₙ = a₁ × (rⁿ − 1) / (r − 1) (for r ≠ 1), which grows very quickly for ratios greater than 1.
How do I find the nth term of a geometric sequence?
Use the formula aₙ = a₁ × r^(n−1), where a₁ is the first term, r is the common ratio, and n is the term number you want.
What is the 5th term of the sequence starting at 2 with a common ratio of 3?
a₅ = 2 × 3⁴ = 2 × 81 = 162.
Geometric Sequence Calculator
A geometric sequence multiplies by the same fixed ratio to get from one term to the next, which means it can grow (or shrink) far more dramatically than an arithmetic sequence over the same number of steps. Enter the first term, the common ratio, and which term number you want, and this calculator finds that term directly along with the sum of all terms up to that point. It's especially useful for modeling compounding processes, where each step scales by a constant factor rather than adding a constant amount.
- In a geometric sequence, each term is found by multiplying the previous term by a constant common ratio, e.g. 2, 6, 18, 54, 162… has a common ratio of 3.
- The nth term formula skips ahead directly: aₙ = a₁ × r^(n−1), avoiding the need to multiply repeatedly.
- The sum of the first n terms uses the formula Sₙ = a₁ × (rⁿ − 1) / (r − 1) (for r ≠ 1), which grows very quickly for ratios greater than 1.
How do I find the nth term of a geometric sequence?
Use the formula aₙ = a₁ × r^(n−1), where a₁ is the first term, r is the common ratio, and n is the term number you want.
What is the 5th term of the sequence starting at 2 with a common ratio of 3?
a₅ = 2 × 3⁴ = 2 × 81 = 162.