Arithmetic series

An arithmetic sequence adds a constant difference; a series adds its terms.

With first term a and common difference d, uₙ = a + (n − 1)d. The sum of n terms is n(a + uₙ)/2. Pairing first and last terms explains the formula: each pair has the same total.

Do not confuse a term number with a term’s value. If the final value is supplied, solve for its index before using it as the number of terms. AQA 8365 specifies linear nth terms, not a dedicated arithmetic-series sum requirement.

Worked example

Sum 5 + 8 + 11 + … + 32.

32 = 5 + (n − 1)3 gives n = 10.
Use S₁₀ = 10(5 + 32)/2.

Answer: 185

Revise first: Quadratic sequences.

Practise arithmetic series

Course mapping

These specification references show where the topic occurs. The questions cover only some parts of each topic.

9FM0 · core · Assumed A-level Mathematics knowledge: Arithmetic series sums
7367 · core · Assumed A-level Mathematics knowledge: Arithmetic series sums
H245 · core · Assumed A-level Mathematics knowledge: Arithmetic series sums
H645 · core · Assumed A-level Mathematics knowledge: Arithmetic series sums
4PM1 · core · 5B: Arithmetic series sums

Next practice: Geometric series.