Sequences, series, and induction proofs (IB Math AA HL)
Updated October 2026
Quick answer: Arithmetic: uₙ = u₁ + (n − 1)d and Sₙ = (n/2)(2u₁ + (n − 1)d). Geometric: uₙ = u₁rⁿ⁻¹ and Sₙ = u₁(1 − rⁿ)/(1 − r), with sum to infinity u₁/(1 − r) when |r| < 1. AA HL combines these with sigma notation, inequalities in n, and proof by induction of sum formulas.
HL AA combines sequences with proof by induction — 'prove that the sum of the first n squares is n(n+1)(2n+1)/6'.
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Sigma notation: Σ from r = 1 to n of (3r + 2) is an arithmetic series with first term 5 and common difference 3. Write out the first few terms if unsure.
A geometric series converges only when |r| < 1. If r is in terms of x, solve −1 < r < 1 for x.
To find the least n with Sₙ above a value, form the inequality, then use logarithms or a table on the GDC. Dividing by the log of a number below 1 reverses the inequality.
Sₙ − Sₙ₋₁ = uₙ, which finds a term from a formula for the sum.
A sum formula such as Σ r² = n(n + 1)(2n + 1)/6 is proved by induction: add the (k + 1)th term to the formula for k terms and simplify to the formula for k + 1.
Worked example
A geometric series has first term 12 and sum to infinity 48. Find the common ratio, and the least number of terms for which the sum is greater than 47.
S∞ = 12/(1 − r) = 48, so 1 − r = 1/4 and r = 3/4 (and |r| < 1, as needed).
A calculator is allowed on AA Paper 2 and Paper 3, not on Paper 1. Keystrokes for the TI-84 Plus CE, TI-Nspire CX II and Casio fx-CG50, with a checked worked example:
When does a geometric series have a sum to infinity?
Only when the common ratio satisfies |r| < 1, so the terms shrink towards 0. The sum is then u₁/(1 − r). If |r| ≥ 1 the partial sums grow without limit or keep oscillating.
Can I use my calculator for sequences, series, and induction proofs?
In HL AA, Paper 1 is non-calculator and the other papers allow a GDC, so learn the exact method as well as the calculator one. The Sequences and sums guide gives the keystrokes for the TI-84 Plus CE, TI-Nspire CX II and Casio fx-CG50.