Arithmetic and geometric sequences

Arithmetic sequence

Each term is obtained by adding a constant common difference (d) to the previous one:

uₙ₊₁ = uₙ + d
General term: uₙ = u₀ + n × d
Sum of the first n terms: Sₙ = n × (u₀ + uₙ₋₁) / 2

Example: 2, 5, 8, 11... (common difference d = 3)

Geometric sequence

Each term is obtained by multiplying the previous one by a constant common ratio (r):

uₙ₊₁ = uₙ × r
General term: uₙ = u₀ × rⁿ
Sum of the first n terms (r ≠ 1): Sₙ = u₀ × (1 − rⁿ) / (1 − r)

Example: 3, 6, 12, 24... (common ratio r = 2)

A worked example

For the arithmetic sequence 4, 7, 10, 13... (u₀=4, d=3), the 10th term (n=9):

u₉ = 4 + 9 × 3 = 31

A real-world application

Simple interest follows an arithmetic progression (the same amount added each period); compound interest follows a geometric progression (the balance is multiplied by the same factor each period).