How this calculator works
This calculator finds the nth term and the sum of the first n terms for two kinds of number sequences. An arithmetic sequence adds the same amount to each term (3, 8, 13, 18, ...); a geometric sequence multiplies each term by the same ratio (1, 2, 4, 8, ...).
For geometric sequences with a common ratio between −1 and 1, the calculator also reports the sum to infinity — the finite total the series approaches as you add infinitely many, ever-shrinking terms, such as 1 + 1/2 + 1/4 + 1/8 + ... approaching 2.
The formulas
Arithmetic nth term: aₙ = a₁ + (n − 1)dArithmetic sum of n terms: Sₙ = n/2 × (2a₁ + (n − 1)d)Geometric nth term: aₙ = a₁ × r^(n − 1)Geometric sum of n terms: Sₙ = a₁ × (1 − rⁿ) ÷ (1 − r), or a₁ × n when r = 1Geometric sum to infinity: S = a₁ ÷ (1 − r), only when |r| < 1When |r| ≥ 1, a geometric series has no finite sum to infinity — the terms never shrink toward zero, so the running total either grows without bound or oscillates forever. This calculator states that plainly instead of showing a number.
Worked example: arithmetic sequence a₁ = 3, d = 5, n = 10
- The sequence is 3, 8, 13, 18, 23, 28, 33, 38, 43, 48.
- 10th term: a₁₀ = 3 + (10 − 1) × 5 = 3 + 45 = 48.
- Sum of 10 terms: S₁₀ = 10/2 × (2×3 + 9×5) = 5 × 51 = 255.
- Check by brute-force addition: 3+8+13+18+23+28+33+38+43+48 = 255, matching the formula.
Frequently asked questions
What is the difference between an arithmetic and a geometric sequence?
An arithmetic sequence has a constant difference between consecutive terms (add d each time). A geometric sequence has a constant ratio between consecutive terms (multiply by r each time). Arithmetic sequences grow linearly; geometric sequences grow (or shrink) exponentially.
When does a geometric series have a sum to infinity?
Only when the common ratio r is strictly between −1 and 1 (|r| < 1). In that case each term is smaller than the last, so the running total converges to a fixed value: a₁ ÷ (1 − r). If |r| ≥ 1, the terms don't shrink, so the sum grows without bound or never settles.
Can the common difference or ratio be negative?
Yes. A negative common difference produces a decreasing arithmetic sequence, and a negative common ratio produces a geometric sequence that alternates sign, such as 2, −4, 8, −16, ...
What if the common ratio is exactly 1?
Every term equals a₁, so the sequence is constant and the sum of n terms is simply a₁ × n. The standard geometric-sum formula would divide by zero at r = 1, so this calculator uses that simpler formula instead.
How is this different from a series?
A sequence is the ordered list of terms itself (3, 8, 13, ...). A series is the sum of those terms (3 + 8 + 13 + ...). This calculator reports both: individual terms and their running sum.