What is my sequence's nth term?

How it works

Terms are numbered from 1. For an arithmetic sequence with first term a₁ and common difference d:

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

For a geometric sequence with first term a₁ and common ratio r:

• nth term: aₙ = a₁ × r^(n − 1) (with r^0 = 1, also when r is 0)
• sum of the first n terms: Sₙ = a₁(1 − rⁿ)/(1 − r) when r is not 1, and Sₙ = n × a₁ when r = 1
• sum to infinity, only when −1 < r < 1: S∞ = a₁/(1 − r)

The numbers are read as the exact decimals typed (0.1 is exactly one tenth), the formulas are worked out in exact fractions, and each result is then shown as the nearest 64-bit floating-point number, to at most 6 decimal places. A term or a sum beyond the largest 64-bit float (about 1.8 × 10^308) has no answer. The first term and the difference are from −10^9 to 10^9, the ratio from −1,000 to 1,000, and n from 1 to 10,000.

The calculator shows:

• nth term (aₙ), Sum of the first n terms (Sₙ), and, for a geometric sequence with −1 < r < 1, Sum to infinity.
• First terms: a₁, a₂, … up to 10 terms (or n terms when n is less than 10), each written as a decimal in full (no exponent, no thousands separators, a plain hyphen for a minus sign) when it needs at most 20 decimal places and at most 30 digits, and otherwise as the shortest text of its nearest 64-bit float in JavaScript's number notation (1e+25, 5e-324), joined by a comma and a space, and followed by a comma, a space and … when n is more than 10.
• Explicit formula, with a₁, d and r written the same way as the terms:
  • arithmetic: a_n = a₁ + d(n - 1), with - |d| for a negative d, + (n - 1) for d = 1, - (n - 1) for d = −1, and just a_n = a₁ for d = 0;
  • geometric: a_n = a₁ × r^(n - 1), with a negative r in brackets ((-3)^(n - 1)) and a₁ × left out when a₁ is 1.
• Recursive formula: a_1 = a₁, a_n = a_(n - 1) + d (or - |d|) for arithmetic, and a_1 = a₁, a_n = r × a_(n - 1) (a negative r in brackets) for geometric.

Worked examples by hand

Arithmetic, a₁ = 2, d = 3, n = 10. The terms are 2, 5, 8, 11, 14, 17, 20, 23, 26, 29. a₁₀ = 2 + 9 × 3 = 29, and S₁₀ = 10 × (2 + 29)/2 = 155.

Geometric, a₁ = 3, r = 2, n = 8. The terms are 3, 6, 12, 24, 48, 96, 192, 384. a₈ = 3 × 2⁷ = 384, and S₈ = 3 × (1 − 2⁸)/(1 − 2) = 3 × 255 = 765.

Geometric, a₁ = 1, r = 0.5, n = 5. The terms are 1, 0.5, 0.25, 0.125, 0.0625, so a₅ = 0.0625 and S₅ = 1 × (1 − 0.03125)/0.5 = 1.9375. Because −1 < 0.5 < 1, the sum to infinity is 1/(1 − 0.5) = 2.

Arithmetic, a₁ = 0.1, d = 0.2, n = 3. 0.1, 0.3, 0.5, and 0.1 + 0.3 + 0.5 = 0.9 exactly.

Geometric, a₁ = 5, r = −3, n = 4. 5, −15, 45, −135, and S₄ = 5 × (1 − 81)/(1 + 3) = −100.

Arithmetic, a₁ = 10, d = −4, n = 6. 10, 6, 2, −2, −6, −10, which add up to 0.