Arithmetic and geometric series · Year 13

In Year 13 you learn: geometric series and sums to infinity.

A series is the sum of the terms of a sequence. Arithmetic series add the same amount each time; geometric series multiply by the same ratio. Their sum formulas answer questions like “how much will I have saved after five years?” in one step.

Key facts

• Arithmetic: uₙ = a + (n − 1)d and Sₙ = n/2 (2a + (n − 1)d) = n/2 (a + l).
• Geometric: uₙ = arⁿ⁻¹ and Sₙ = a(1 − rⁿ)/(1 − r).
• If −1 < r < 1, a geometric series converges, with sum to infinity S∞ = a/(1 − r).
• Sigma notation: Σ uᵣ from r = 1 to n means u₁ + u₂ + … + uₙ.
• a is the first term, d the common difference, r the common ratio and l the last term.

Worked examples

Other years

• Arithmetic and geometric series · Year 12