Sequence Calculator

Choose the sequence type, enter the first term and the common difference or ratio, and pick which term you need.

Sequence

Term 10 (aₙ)39
Sum of the first 10 terms
210
First terms
3, 7, 11, 15, 19, 23, 27, 31, 35, 39

Working

  1. aₙ = a₁ + (n − 1)d = 3 + 9 × 4 = 39
  2. Sₙ = n/2 × (a₁ + aₙ) = 10/2 × (3 + 39) = 210

Formulas

Arithmetic sequence

aₙ = a₁ + (n − 1)d

Sₙ = n ÷ 2 × (a₁ + aₙ)

Geometric sequence

aₙ = a₁ × rⁿ⁻¹

Sₙ = a₁(1 − rⁿ) ÷ (1 − r), r ≠ 1

Worked example

Arithmetic: 3, 7, 11, … — find a₁₀ and S₁₀

  1. a₁₀ = 3 + 9 × 4 = 39
  2. S₁₀ = 10 ÷ 2 × (3 + 39) = 5 × 42 = 210

The 10th term is 39 and the first ten terms add up to 210.

Recognising the type

If the difference between consecutive terms is constant, the sequence is arithmetic. If the ratio is constant, it is geometric. Savings with a fixed monthly deposit grow arithmetically; a quantity that doubles each period grows geometrically.