How to use this calculator
- Enter or choose sequence type.
- Enter or choose first term (a₁).
- Enter or choose term number (n).
- The nth term updates as you type. Use Copy result or Share to keep it.
Method
Arithmetic: aₙ = a₁ + (n − 1)d, Sₙ = n(2a₁ + (n − 1)d) ÷ 2; Geometric: aₙ = a₁·rⁿ⁻¹, Sₙ = a₁(1 − rⁿ) ÷ (1 − r), S∞ = a₁ ÷ (1 − r) for |r| < 1; Fibonacci: Fₙ = Fₙ₋₁ + Fₙ₋₂
- a₁
- first term
- d
- common difference
- r
- common ratio
- n
- position of the term
Worked example
- a₁₀ = 2 + (10 − 1) × 3 = 2 + 27 = 29.
- S₁₀ = 10 × (2 × 2 + 9 × 3) ÷ 2 = 10 × 31 ÷ 2 = 155.
- Check: the first and last terms average (2 + 29) ÷ 2 = 15.5, and 15.5 × 10 = 155.
In 2, 5, 8, …, the 10th term is 29 and the first ten terms add up to 155.
What the result means
An arithmetic sequence grows by a fixed amount (like simple interest); a geometric one grows by a fixed factor (like compound interest), so it eventually outruns any arithmetic sequence when r > 1.
Assumptions and limits
- The first term is term number 1. Fibonacci starts F₁ = F₂ = 1.
- Results are shown to 15 significant digits; very large geometric terms are approximate beyond that.
Questions
When does a geometric series have a finite sum to infinity?
Only when the ratio is between −1 and 1. Then the terms shrink fast enough that the total approaches a₁ ÷ (1 − r); for example 1 + ½ + ¼ + … = 2.
How do I find the common difference?
Subtract any term from the next one. If the differences are not constant but the ratios are, the sequence is geometric.