Number Sequence Calculator

A sequence lists numbers in a fixed order according to a rule, and each number in it is called a term. Sequences can be finite or go on forever. Some are defined by a formula for the nth term, like the even numbers 2n, and others are defined recursively, where each term is computed from earlier ones, like the Fibonacci numbers. The sum of the terms of a sequence is called a series.

Finding the rule behind a list of numbers is a common puzzle, and a good first test is to look at the differences between neighboring terms. Constant differences mean an arithmetic sequence. If the differences are not constant but the ratios are, the sequence is geometric. If the differences themselves have constant differences, the nth term follows a quadratic formula, as in the square numbers 1, 4, 9, 16, 25. And if each term is the sum of the previous two, it is a Fibonacci-type sequence.

A short list never determines a sequence completely: many different rules produce 1, 2, 4 and then diverge. 1, 2, 4, 8, 16 doubles, but 1, 2, 4, 7, 11 adds increasing amounts. That is why the calculators above ask for the rule itself, the first term together with the common difference or ratio, rather than guessing from a few terms.

Sequences appear throughout finance and science. Simple interest grows in an arithmetic sequence, compound interest and population growth in a geometric one, and the Fibonacci numbers turn up in the spirals of sunflowers and pine cones. Each of the three calculators also reports the sum of the terms, which answers questions such as the total of a series of savings deposits or the number of seats in a theater whose rows grow by a fixed amount.

Arithmetic Sequence

In an arithmetic sequence, each term is the previous term plus a fixed number called the common difference, d. The default example starts at 2 and adds 5 each time: 2, 7, 12, 17, 22, and so on. The common difference can be negative, making the sequence decrease, or a fraction.

The nth term is the first term plus n − 1 steps of size d. For the default, the 20th term is 2 + 19 × 5 = 97. The sum of the first n terms equals the number of terms times the average of the first and last term, a shortcut the young Carl Friedrich Gauss is said to have used to add 1 through 100 instantly: 100 × (1 + 100) ÷ 2 = 5,050. The sum of the first 20 terms of the default sequence is 20 × (2 + 97) ÷ 2 = 990.

aₙ = a₁ + (n − 1)d, Sₙ = n(a₁ + aₙ) ÷ 2

a₁

the first term

d

the common difference

n

the position of the term

Sₙ

the sum of the first n terms

Geometric Sequence

In a geometric sequence, each term is the previous term multiplied by a fixed common ratio r. Starting at 2 with r = 5 gives 2, 10, 50, 250 and so on. The nth term is aₙ = a₁ × rⁿ⁻¹, so the 12th term of the default is 2 × 5¹¹ = 97,656,250.

The sum of the first n terms is Sₙ = a₁(1 − rⁿ) ÷ (1 − r) for r ≠ 1, which gives 122,070,312 for the default. When the ratio is between −1 and 1, the terms shrink toward zero and the infinite sum converges to a₁ ÷ (1 − r); 1 + 1/2 + 1/4 + 1/8 + … adds up to exactly 2.

Fibonacci Sequence

The Fibonacci sequence begins 0, 1, and every later term is the sum of the two before it: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55. It is named after Leonardo of Pisa, known as Fibonacci, who used it in 1202 to model a growing rabbit population, though Indian mathematicians had described it centuries earlier. The ratio of consecutive terms approaches the golden ratio φ = (1 + √5) ÷ 2 ≈ 1.618, and Binet's formula Fₙ = (φⁿ − (1 − φ)ⁿ) ÷ √5 gives any term directly. With F₀ = 0, the 10th Fibonacci number is 55.

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