Number Sequence Calculator
The nth term and the sum of an arithmetic, geometric or Fibonacci sequence, the infinite sum where it exists, and the rule behind a few terms you type.
Result
The nth term: 48
Fibonacci numbers are exact to any size; other terms to the precision of ordinary numbers.
- Sum of the first n terms
- 255
- The first terms
- 3, 8, 13, 18, 23, 28, 33, 38, 43, 48
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Results follow the mathematical operation and values shown. Check that you selected the intended operation, convention and dataset type before relying on the result. Read the full disclaimer
How to use the Number Sequence Calculator
- Choose arithmetic, geometric or Fibonacci, or enter a few terms and let the page find the rule.
- Enter the first term and the common difference or ratio, and the term number n.
- Read the nth term, the sum of the first n terms, and for a shrinking geometric sequence, the sum of all its terms.
The formulas
arithmetic aₙ = a + (n − 1)d Sₙ = n(2a + (n − 1)d) ÷ 2geometric aₙ = a × rⁿ⁻¹ Sₙ = a(1 − rⁿ) ÷ (1 − r)infinite S = a ÷ (1 − r), only when −1 < r < 1Fibonacci Fₙ = Fₙ₋₁ + Fₙ₋₂, starting 1, 1The geometric sum formula divides by 1 − r, which is zero when r = 1. Then every term equals a, and the sum of n of them is simply n × a; the page handles that case separately.
The sum of Fibonacci numbers
The first n Fibonacci numbers add up to one less than the number two places further on: 1 + 1 + 2 + … + 55 = 144 − 1 = 143.
With your numbers
- aₙ = a + (n − 1)d
Quick reference
Fibonacci numbers are exact to any size; other terms to the precision of ordinary numbers.
| Example | Result |
|---|---|
| Arithmetic 3, +5, 10th term | The nth term: 48 |
| Geometric 2, ×3, 5th term | The nth term: 162 |
| Geometric 1, ×1 | The nth term: 1 |
| Geometric 1, ×0.5, 10 terms | The nth term: 0.001953 |
| Fibonacci, 10th | The nth Fibonacci number, exactly: 55 |
| Fibonacci, 100th | The nth Fibonacci number, exactly: 354224848179261915075 |
| Detect 2, 6, 18, 54 | The nth term: 486 |
Common questions about the Number Sequence Calculator
- What is the difference between arithmetic and geometric sequences?
- An arithmetic sequence adds the same amount each time: 3, 8, 13, 18. A geometric sequence multiplies by the same amount: 2, 6, 18, 54. Arithmetic sequences grow in a straight line; geometric ones grow, or shrink, exponentially.
- How can an infinite sum be finite?
- When each term is a fixed fraction of the one before, and that fraction is less than one in size. 1 + 1/2 + 1/4 + 1/8 + … gets ever closer to 2 without passing it, and the sum of all the terms is exactly 2. With a ratio of 1 or more, the terms never shrink and the sum grows without limit, so the page gives no number.
- How big do Fibonacci numbers get?
- Very big, very fast: each is roughly 1.618 times the one before. The 100th is 354,224,848,179,261,915,075, twenty-one digits. The page works them out with exact whole-number arithmetic, so every digit is right, where ordinary computer numbers go wrong after the 78th.
- What if my terms fit neither rule?
- The page checks for a constant difference, then a constant ratio. If neither holds, it says so rather than guessing: many sequences, such as square numbers, follow other rules.