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IB Maths: Analysis & Approaches revision notes Group 5 · Maths · SL / HL
Sequences, series and logarithms show up in almost every paper. Learn four formulas well and you unlock a lot of easy marks.
Course topics Number and algebra · Sequences, series, logs, binomial theorem, proof · 16 subtopics Functions · Domain, range, inverses, transformations, quadratics, exponentials · 16 subtopics Geometry and trigonometry · 3D shapes, sine and cosine rules, radians, identities · 18 subtopics Statistics and probability · Data, regression, probability, binomial and normal distributions · 14 subtopics Calculus · Limits, differentiation, integration, kinematics, optimisation · 19 subtopics Free lesson: Number and algebra Arithmetic sequence You add the same number (d) each time: 3, 7, 11, 15… Geometric sequence You multiply by the same number (r) each time: 5, 10, 20, 40… Logarithm A log answers “what power?”: log₂ 8 = 3 because 2³ = 8 Arithmetic (common difference d)
uₙ = u₁ + (n − 1)d
Sₙ = n/2 · (2u₁ + (n − 1)d) = n/2 · (u₁ + uₙ)
Geometric (common ratio r)
uₙ = u₁ · rⁿ⁻¹
Sₙ = u₁(1 − rⁿ) / (1 − r) r ≠ 1
S∞ = u₁ / (1 − r) only if |r| < 1
Logarithms
logₐ x = y ⇔ aʸ = x
log(xy) = log x + log y log(x/y) = log x − log y
log(xⁿ) = n · log x logₐ x = log x / log a
Worked example An arithmetic sequence has u₃ = 11 and u₇ = 27. Find the sum of the first 20 terms.
Write both terms with the formula: u₁ + 2d = 11 and u₁ + 6d = 27. Subtract: 4d = 16, so d = 4. Substitute back: u₁ = 11 − 8 = 3. Sum: S₂₀ = 20/2 · (2·3 + 19·4) = 10 · 82 = 820 . Common mistakes Using n instead of (n − 1) in the term formula. Using S∞ when |r| ≥ 1 — the series has no sum. Forgetting that a negative ratio makes terms alternate in sign. Writing log(x + y) = log x + log y. That law does not exist. Quick check A geometric sequence has u₁ = 5 and r = 2. What is u₆? 160 . u₆ = 5 · 2⁵ = 5 · 32 = 160. The power is n − 1 = 5, not 6.
What is log₂ 32? 5 . 2⁵ = 32, so log₂ 32 = 5.
Find the sum to infinity of 12 + 6 + 3 + … 24 . r = 0.5, which is between −1 and 1. S∞ = 12 / (1 − 0.5) = 24.
An arithmetic sequence starts at 7 with d = −3. What is u₁₀? −20 . u₁₀ = 7 + 9 · (−3) = 7 − 27 = −20.
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FAQ What topics are in IB Maths: Analysis & Approaches? IB Maths: Analysis & Approaches (SL / HL) is organised into: Number and algebra; Functions; Geometry and trigonometry; Statistics and probability; Calculus.
Are these IB Maths: Analysis & Approaches notes free? Yes. The topic pages, key terms and the first lesson are free. The full lessons, interactive diagrams and exam-style practice papers with mark schemes are in the FortyFive app, which you can start for free.
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