Nth Term Sequence Calculator
Find the nth term and the running sum of an arithmetic or geometric sequence from its first term and common difference or ratio.
How it works
Arithmetic: aₙ = a₁ + (n − 1)d · Sₙ = n⁄2 (2a₁ + (n − 1)d) · Geometric: aₙ = a₁·rⁿ⁻¹ · Sₙ = a₁(rⁿ − 1)⁄(r − 1)
An arithmetic sequence adds a fixed amount (the common difference) each step, so its terms grow linearly. A geometric sequence multiplies by a fixed ratio each step, so its terms grow exponentially. The closed-form nth-term formulas let you jump straight to any position without listing every term, and the sum formulas total the whole run at once.
Worked example
For the arithmetic sequence starting at 2 with common difference 3, the 10th term is 2 + (10 − 1)×3 = 2 + 27 = 29. The sum of the first ten terms is 10⁄2 × (2×2 + 9×3) = 5 × 31 = 155.
Sources: Arithmetic progression — Wikipedia
Frequently asked questions
What is the difference between arithmetic and geometric?
Arithmetic sequences add the same number every step (2, 5, 8, 11…), giving straight-line growth. Geometric sequences multiply by the same number every step (2, 6, 18, 54…), giving exponential growth.
Can the common difference or ratio be negative or a fraction?
Yes. A negative difference makes an arithmetic sequence decrease; a ratio between 0 and 1 makes a geometric sequence shrink toward zero. This tool handles all of these.
Why is there a cap on n?
Very large term numbers — especially in geometric sequences — produce numbers too big to represent accurately, so n is limited to 10,000 to keep results reliable.
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This calculator is for educational purposes. Double-check important results before acting on them.