Arithmetic and geometric sequences
Give the first term, the difference or ratio, and the term number. You get that term, the sum and the start of the sequence.
Result
- First terms
- 3; 7; 11; 15; 19; 23; 27; 31; 35; 39
aₙ = a₁ + (n − 1) · d
Sₙ = (a₁ + aₙ) · n / 2
Arithmetic sequences
Each term is the previous one plus a constant difference d: 3, 7, 11, 15… The general term is aₙ = a₁ + (n − 1)·d and the sum of n terms is Sₙ = (a₁ + aₙ)·n / 2. Gauss supposedly used this to add 1 + 2 + … + 100 = 5050 in his head.
Geometric sequences
Each term is the previous one times the ratio r: 2, 6, 18, 54… The general term is aₙ = a₁·rⁿ⁻¹ and the sum is Sₙ = a₁·(1 − rⁿ)/(1 − r). When |r| < 1 the terms shrink to zero and the sum of all terms is finite: S = a₁/(1 − r).
Common questions
How do I tell if a sequence is arithmetic?
Work out the differences between consecutive terms. If they are constant, it is arithmetic; if the ratios are constant, it is geometric.