Sequences and series

Arithmetic adds, geometric multiplies

Arithmetic: aₙ = a₁ + (n−1)d. Geometric: aₙ = a₁·rⁿ⁻¹. Work out which by checking whether consecutive terms differ by the same amount or the same factor.

Sums

Arithmetic: n(a₁ + aₙ)/2, the average term times how many. Geometric: a₁(1 − rⁿ)/(1 − r).

Infinite geometric series

Only converges when |r| < 1, and then the sum is a₁/(1 − r). If |r| ≥ 1 it grows without limit and there is no answer to give.

Off-by-one

The n−1 in the arithmetic formula catches everyone. The first term uses n = 1, so nothing has been added yet.