How to use the Sequence Calculator
- Choose arithmetic if the terms change by a fixed amount, geometric if they change by a fixed multiple.
- Enter the first term a₁.
- Enter the common difference d for arithmetic, or the common ratio r for geometric.
- Enter the term number n you want to reach.
- Read the nth term and the sum of all terms up to it; the infinite sum appears only when |r| < 1.
How the calculation works
An arithmetic sequence adds the same amount each step, so the nth term is the first term plus (n − 1) copies of the difference. Its partial sum has an elegant shortcut: pair the first term with the last, the second with the second-last, and every pair has the same total. That gives the sum as n/2 times the sum of the first and last terms — the trick attributed to a young Gauss summing 1 to 100.
A geometric sequence multiplies by the same factor each step, so the nth term is a₁ multiplied by r to the power (n − 1). The partial sum comes from a telescoping argument: subtract r times the sum from the sum itself and everything cancels except the ends, leaving a₁(1 − rⁿ)/(1 − r). The special case r = 1 must be handled separately, since it would divide by zero, and there the sum is simply n copies of a₁.
Convergence is where geometric series become genuinely useful. When |r| < 1 the terms shrink towards zero fast enough that the infinite sum settles on a finite value, a₁/(1 − r). This is the mathematics behind the present value of a perpetuity, the total distance travelled by a bouncing ball, the money multiplier in economics, and the geometric distribution's expected value. When |r| ≥ 1 the sum grows without bound and the infinite total is meaningless.
Arithmetic: aₙ = a₁ + (n−1)d, Sₙ = n/2·(2a₁ + (n−1)d). Geometric: aₙ = a₁·r^(n−1), Sₙ = a₁(1 − rⁿ)/(1 − r), S∞ = a₁/(1 − r) for |r| < 1Source: Standard sequence and series results; see OpenStax Algebra and Trigonometry, Chapter 11.
Worked example
A savings plan starts at $200 a month and increases by $25 each month. What is deposited in month 24, and in total over two years?
- Arithmetic with a₁ = 200, d = 25, n = 24.
- Month 24 deposit: 200 + 23 × 25 = 200 + 575 = $775.
- Sum: 24/2 × (2 × 200 + 23 × 25) = 12 × (400 + 575) = 12 × 975.
- Total = $11,700, ignoring any interest earned.
$775 in the final month and $11,700 contributed across the two years, before interest.
Frequently asked questions
How do I tell arithmetic from geometric?+
Subtract consecutive terms: a constant difference means arithmetic. Divide consecutive terms: a constant ratio means geometric. If neither is constant, it is some other kind of sequence.
Why does the infinite sum only exist sometimes?+
It requires the terms to shrink towards zero quickly. With |r| ≥ 1 each term is as large as or larger than the last, so the running total never settles.
Is the Fibonacci sequence covered here?+
No. Fibonacci is defined by a recurrence on the two previous terms, not by a fixed difference or ratio, so it needs a different formula.
Can the common ratio be negative?+
Yes. The terms then alternate in sign, and the series still converges as long as the absolute value of r is below one.
Last reviewed August 31, 2026. We review this page whenever the underlying formula, tax year, published rate or standard changes.