Arithmetic Sequences & Series
nth term
uₙ = u₁ + (n − 1)d Sum of n terms
Sₙ = n/2 × (2u₁ + (n − 1)d) = n/2 × (u₁ + uₙ) IB Diploma Programme Mathematics AI SL / HL
📊 IB DP Mathematics AI Formula Sheet 2026
Key formulas and relationships for IB DP Mathematics Applications & Interpretations SL and HL — focused on real-world modelling, statistics, and technology-assisted problem solving.
Number & Algebra Functions Geometry & Trig Statistics & Probability Calculus (HL)
Our formula sheets are free to download — save this one as PDF for offline revision.
Aligned with the latest 2026 syllabus and board specifications. This sheet is prepared to match your exam board’s official specifications for the 2026 exam series.
IB Maths AI — Real-World Applications of Mathematics
IB DP Mathematics: Applications & Interpretations (AI) focuses on the use of mathematics to model and solve real-world problems. Unlike Maths AA (Analysis & Approaches), AI places greater emphasis on statistics, technology, and practical applications. This formula sheet covers all five topic areas for both SL and HL, with HL-only content clearly marked.
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Financial mathematics and sequences
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Functions including logistic and piecewise models
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3D geometry, trigonometry, and Voronoi diagrams
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Statistics, probability distributions, and hypothesis testing
Number & Algebra
AI Maths emphasises financial applications of arithmetic and geometric sequences.
Geometric Sequences & Series
nth term
uₙ = u₁ × r^(n−1) Sum of n terms
Sₙ = u₁(rⁿ − 1) / (r − 1), r ≠ 1 Sum to infinity
S∞ = u₁ / (1 − r), |r| < 1 Financial Mathematics
Key to AI Maths — compound interest, annuities, loans.
Compound interest
FV = PV × (1 + r/k)^(kn) FV = future value, PV = present value, r = annual interest rate, k = compounding periods per year, n = number of years.
Loan repayment (GDC)
Use TVM Solver on GDC for annuity calculations Logarithms (HL)
Change of base
logₐ b = ln b / ln a Laws
log(xy) = log x + log y | log(x/y) = log x − log y | log xⁿ = n log x Functions
AI focuses on interpreting and applying functions in real-world contexts.
Linear Functions
Gradient
m = (y₂ − y₁) / (x₂ − x₁) Equation
y = mx + c or y − y₁ = m(x − x₁) Quadratic Functions
Standard form
y = ax² + bx + c Vertex form
y = a(x − h)² + k, vertex at (h, k) Quadratic formula
x = (−b ± √(b² − 4ac)) / 2a Discriminant
Δ = b² − 4ac (>0: two real roots; =0: one; <0: no real roots) Exponential & Logarithmic Functions
Exponential growth/decay
y = a × bˣ or y = a × e^(kx) b > 1: growth; 0 < b < 1: decay. Used for population growth, radioactive decay, compound interest.
Logistic Function (AI specific)
Used to model limited growth (e.g., population with carrying capacity).
y = L / (1 + Ce^(−kx)) L = carrying capacity (maximum value). As x → ∞, y → L. S-shaped (sigmoid) curve.
Sinusoidal Functions
y = a sin(b(x − c)) + d a = amplitude; period = 2π/b; c = horizontal shift; d = vertical shift (midline).
Geometry & Trigonometry
AI Maths includes 3D geometry and real-world spatial problems.
Trigonometric Ratios (right-angled triangles)
sin θ = opposite/hypotenuse | cos θ = adjacent/hypotenuse | tan θ = opposite/adjacent Sine Rule & Cosine Rule (non-right triangles)
Sine rule
a/sin A = b/sin B = c/sin C Cosine rule (find side)
a² = b² + c² − 2bc cos A Cosine rule (find angle)
cos A = (b² + c² − a²) / 2bc Area of Triangle
Standard
Area = ½ × base × height Using two sides and included angle
Area = ½ab sin C 3D Geometry
Distance between two 3D points
d = √((x₂−x₁)² + (y₂−y₁)² + (z₂−z₁)²) Volume of sphere
V = (4/3)πr³ Surface area of sphere
A = 4πr² Voronoi Diagrams (AI specific)
Partition a plane into regions based on proximity to a set of seed points.
Nearest neighbour principle: every point in a region is closer to its seed than to any other seed. Used for facility allocation problems.
Statistics & Probability
Statistics is a major component of AI Maths — know all distributions and test conditions.
Descriptive Statistics
Mean
x̄ = Σx / n (or use GDC) Standard deviation (population)
σ = √(Σ(x − x̄)² / n) IQR = Q3 − Q1. Box plots and outliers: values beyond Q1 − 1.5×IQR or Q3 + 1.5×IQR.
Correlation & Regression
Pearson's r
Value between −1 and +1; |r| > 0.7 is considered strong correlation Least squares regression
y = ax + b (use GDC to find a and b) Only interpret regression line within the data range (interpolation). Extrapolation is unreliable.
Normal Distribution
Notation: X ~ N(μ, σ²) Standardising
z = (x − μ) / σ 68–95–99.7 rule: 68% within ±1σ, 95% within ±2σ, 99.7% within ±3σ.
Binomial Distribution
X ~ B(n, p): P(X = r) = C(n,r) × pʳ × (1−p)^(n−r) Mean: μ = np | Variance: σ² = np(1−p). Use GDC for cumulative probabilities.
Poisson Distribution (HL)
X ~ Po(λ): P(X = x) = (e^(−λ) × λˣ) / x! Mean = Variance = λ. Used for rare events in a fixed interval of time or space.
Hypothesis Testing
Chi-squared test
χ² = Σ(O − E)²/E | Goodness of fit or test of independence t-test
Test whether population mean equals a given value (one-sample) or compare two means Compare test statistic to critical value, or compare p-value to significance level α. Reject H₀ if p < α.
Calculus (HL)
HL only — calculus in AI Maths focuses on applied differentiation and integration.
Differentiation
Power rule
d/dx(xⁿ) = nxⁿ⁻¹ Chain rule
d/dx[f(g(x))] = f'(g(x)) × g'(x) Product rule
d/dx[uv] = u'v + uv' Quotient rule
d/dx[u/v] = (u'v − uv') / v² Applications of Differentiation
Gradient at a point
Substitute x-value into dy/dx Turning points
Set dy/dx = 0 → solve → test with d²y/dx²: positive = min, negative = max Integration
Power rule
∫xⁿ dx = xⁿ⁺¹/(n+1) + c, n ≠ −1 Definite integral (area)
Area = ∫ₐᵇ f(x) dx For area between curve and x-axis: if f(x) < 0 in [a,b], take absolute value of integral.
How to Use This Formula Sheet
Boost your Cambridge exam confidence with these proven study strategies from our tutoring experts.
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Master Your GDC
AI Maths is designed to be used with a GDC. Know how to use statistical functions (1-Var Stats, LinReg, distribution calculations, hypothesis tests) fluently.
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Focus on Statistics
Statistics & Probability is the largest topic in AI Maths. Spend extra time on hypothesis testing, normal distribution, and regression interpretation.
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Link to Real Contexts
AI Maths questions are set in real-world contexts. Practise interpreting your mathematical answers in the context of the scenario described.
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Show Intermediate Steps
Even with a GDC, always write down the formula used, the values substituted, and any intermediate steps. Method marks are available.
Formula Sheet FAQ
Quick answers about this free PDF and how to use it for exam revision and active recall.
Is the IB DP Mathematics AI Formula Sheet 2026 free to download as a PDF?
Yes. This Tutopiya formula sheet is free to use and you can download it as a PDF from this page for offline revision. There is no payment or account required for the PDF download.
What Mathematics topics and equations does this formula sheet cover?
This page groups key Mathematics formulas in one place for revision. Complete formula sheet for IB DP Mathematics Applications & Interpretations (AI) SL and HL 2026. Covers Number & Algebra, Functions, Geometry, Statistics & Probability, and Calculus. Always cross-check with your official syllabus and past papers for your exam session.
Can I use this instead of the official exam formula booklet in the exam?
No. In the exam you must follow only what your exam board allows in the hall—usually the official formula booklet or data sheet where provided. This page is a revision and teaching aid, not a replacement for board-issued materials.
Who is this formula sheet for (Post-Secondary)?
It is written for students preparing for assessments at Post-Secondary in Mathematics, including classroom revision, homework support, and independent study. Teachers and tutors can also share it as a quick reference.
How should I revise with this formula sheet?
Work through past paper questions, quote the correct formula before substituting values, and check units and notation every time. Pair this sheet with timed practice and mark schemes so you see how examiners expect working to be set out.
Where can I get more help with Mathematics revision?
Explore Tutopiya’s study tools, past paper finder, and revision checklists linked from our tools hub, or book a trial lesson with a subject specialist for personalised support alongside this formula reference.
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This formula sheet aligns with the IB DP Mathematics: Applications & Interpretations (AI) SL and HL syllabus content.
HL-only content is marked clearly. Always verify GDC output by checking whether the answer is physically reasonable in context.