The Number Sequence Calculator finds the nth term and the sum of arithmetic and geometric sequences, and the nth Fibonacci number.
The 20th number is 97
| 20th term | 97 |
| Sum of the first 20 terms | 990 |
| Sequence | 2, 7, 12, 17, 22, 27, 32, 37, 42, 47, 52, 57, … |
The 12th number is 97656250
| 12th term | 97656250 |
| Sum of the first 12 terms | 122070312 |
| Sequence | 2, 10, 50, 250, 1250, 6250, 31250, 156250, 781250, 3906250, … |
F(10) = 55
| F(10) | 55 |
| Sequence | 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55 |
A sequence is an ordered list of numbers that follows a rule. Sequences appear in mathematics, finance (loan payments and savings), computer science and nature. The three most common types are arithmetic, geometric and Fibonacci.
Each term adds a fixed common difference d: aₙ = a₁ + (n − 1)d. The sum of the first n terms is n(a₁ + aₙ)/2. Example: starting at 2 with d = 5 gives 2, 7, 12, 17, …; the 20th term is 2 + 19 × 5 = 97 and the sum of 20 terms is 20 × (2 + 97) / 2 = 990.
Each term multiplies by a common ratio r: aₙ = a₁ × rⁿ⁻¹. The sum is a₁(1 − rⁿ)/(1 − r) for r ≠ 1. Example: 2, 10, 50, 250, …; the 12th term is 2 × 5¹¹ = 97,656,250. Compound interest is a geometric sequence.
Each term is the sum of the two before: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, … The ratio of consecutive terms approaches the golden ratio φ ≈ 1.618. Fibonacci numbers show up in plant spirals, pine cones and many algorithms.
Can the difference or ratio be negative? Yes; a negative ratio makes terms alternate in sign.
What if the ratio is between −1 and 1? An infinite geometric series then converges to a₁ ÷ (1 − r).
Is F(0) = 0? Yes, by the usual convention.