Arithmetic and Geometric Sequence Calculator

Arithmetic sequences

In an arithmetic sequence each term is found by adding the same number to the one before. That number is the common difference (d): in 3, 7, 11, 15 it is 4.

• nth term: aₙ = a₁ + (n − 1) · d
• Sum of the first n terms: Sₙ = n · (a₁ + aₙ) / 2

Example: with a₁ = 3 and d = 4 the tenth term is 3 + 9 · 4 = 39 and the first ten terms add up to 10 · (3 + 39) / 2 = 210. The same formula gives the sum of the numbers from 1 to 100 as 100 · 101 / 2 = 5050.

Geometric sequences

In a geometric sequence each term is found by multiplying the one before by the same number. That number is the common ratio (r): in 2, 6, 18, 54 it is 3.

• nth term: aₙ = a₁ · rⁿ⁻¹
• Sum of the first n terms: Sₙ = a₁ · (1 − rⁿ) / (1 − r)
• Infinite sum: if |r| < 1 then S∞ = a₁ / (1 − r)

Example: for 1, 1/2, 1/4 and so on, the first ten terms add up to 1023/512 and the infinite sum is 1 / (1 − 1/2) = 2. When the ratio is not smaller than 1 in size the terms do not shrink and there is no infinite sum.

Finding a sequence from two terms

If you know any two terms and their positions, the tool finds the common difference or ratio and the first term. For an arithmetic sequence with a₃ = 7 and a₇ = 19 you get d = (19 − 7) / 4 = 3 and a₁ = 1. For a geometric sequence an even gap between the positions allows two ratios: a₁ = 2 and a₃ = 18 give r = 3 or r = −3.

Position of a term

You can also ask which term of the sequence a value is. In 3, 7, 11 the value 79 is the 20th term. The value 20 is not in the sequence at all. The tool says so and gives the nearest terms.

Limits

The tool handles arithmetic and geometric sequences only. Recursive sequences such as Fibonacci, sequences with a polynomial general term and the harmonic series are not covered. Terms are numbered starting from 1.