Topic 1 · Number & Algebra
Arithmetic Sequences
u_n = u_1 + (n-1)d
S_n = n/2 · (2u_1 + (n-1)d)
u₁ = first term
d = common diff
Example
Find S₂₀ where u₁ = 3, d = 7
S₂₀ = 10(6 + 133) = 1390
Topic 1 · Number & Algebra
Geometric Sequences
u_n = u_1 · r^(n-1)
S_n = u_1(r^n − 1)/(r − 1), r ≠ 1
r = common ratio
|r| < 1 → S∞
Example
Find S₆ where u₁ = 2, r = 3
S₆ = 2(729−1)/2 = 728
Topic 2 · Functions
Exponential Growth & Decay
P(t) = P₀ · (1 + r)^t
Decay: P(t) = P₀ · e^(−kt)
r = growth rate
k = decay constant
Example
P₀ = 5000, r = 4%, t = 10 years
P = 5000(1.04)¹⁰ ≈ 7401
Topic 2 · Functions
Logarithms
log_a(xy) = log_a x + log_a y
log_a(x^n) = n · log_a x
log_b x = ln x / ln b
Change of base
Domain: x > 0
Example
Solve log₂(x) + log₂(x−2) = 3
x(x−2) = 8 → x² − 2x − 8 = 0 → x = 4
Topic 4 · Statistics & Probability
Normal Distribution
X ~ N(μ, σ²)
z = (x − μ) / σ
P(X < a) from GDC / z-table
68–95–99.7 rule
Standardise with z
Example
X~N(70,10²), P(X > 85) = ?
z = 1.5 → P ≈ 0.0668
Topic 4 · Statistics & Probability
Binomial Distribution
X ~ B(n, p)
P(X = k) = C(n,k) · p^k · (1−p)^(n−k)
E(X) = np, Var(X) = np(1−p)
Fixed n trials
Independent events
Example
X~B(10, 0.3), P(X = 3) = ?
C(10,3)·(0.3)³·(0.7)⁷ ≈ 0.2668
Topic 4 · Statistics & Probability
Regression & Correlation
ŷ = ax + b (regression line)
r = correlation coefficient
−1 ≤ r ≤ 1
r > 0.9 strong
Causation ≠ correlation
Example
Data: (1,2.1),(2,3.9),(3,6.2),(4,7.8),(5,10.1)
ŷ ≈ 1.99x + 0.05, r ≈ 0.999
Topic 5 · Calculus
Differentiation
d/dx(xⁿ) = nxⁿ⁻¹
Product: (uv)' = u'v + uv'
Chain: d/dx[f(g)] = f'(g)·g'
f'(x)=0 → stationary
f''<0 → maximum
Example
f(x) = x³ − 6x² + 9x + 2, find stationary points
f'(x) = 3x²−12x+9 = 0 → x = 1, 3
Topic 5 · Calculus
Integration
∫xⁿ dx = xⁿ⁺¹/(n+1) + C
Area = ∫[a to b] |f(x)| dx
Definite integral
Area always positive
Example
Area between y = x²−4 and x-axis, from x = −2 to 2
Area = ∫₋₂² (4−x²) dx = 32/3
Topic 4 · Statistics & Probability
Conditional Probability
P(A|B) = P(A∩B) / P(B)
P(B) = P(B|A)P(A) + P(B|A')P(A')
Bayes' theorem
Tree diagrams
Example
P(A)=0.6, P(B|A)=0.7, P(B|A')=0.3 → P(A|B)?
P(B)=0.54 → P(A|B) = 0.42/0.54 ≈ 0.778
Topic 1 · Number & Algebra
Matrices & Systems of Equations
AX = B → X = A⁻¹B
det(A) = ad − bc
det ≠ 0 → unique sol
Row reduction
Example
Solve: 2x+3y=8, 5x+7y=19
det = −1 → x = 1, y = 2
Topic 3 · Geometry & Trigonometry
Vectors & Dot Product
a · b = |a||b|cos θ
|a| = √(a₁² + a₂² + a₃²)
Perpendicular: a·b=0
Parallel: a = kb
Example
a=(3,4,0), b=(1,0,0): find angle θ
cos θ = 3/5 → θ = 53.1°
Topic 1 · Financial Mathematics
Loan Repayment (Amortisation)
PMT = PV · i / (1−(1+i)⁻ⁿ)
i = annual rate / compounding periods
GDC: TVM solver
i = r/12 monthly
Example
Loan $10 000, 5% annual, 12 monthly payments
PMT ≈ $856.07/month
Topic 2 · Functions
Sinusoidal Functions
f(x) = A sin(Bx − C) + D
Period = 2π/|B|
Amplitude = |A|, Midline = D
Range: [D−A, D+A]
Phase shift = C/B
Example
f(x) = 3sin(2x − π/3) + 1
A=3, Period=π, Range=[−2, 4]
Topic 4 · Statistics & Probability
Chi-squared Test of Independence
χ² = Σ (O−E)² / E
E = (row total × col total) / grand total
df = (r−1)(c−1)
H₀: independent
p < α → reject H₀
Example
2×2 table: O=[[20,30],[25,25]]
χ² ≈ 0.65, p ≈ 0.42 → fail to reject H₀
Topic 5 · Calculus
Separable Differential Equations
dy/dx = f(x)g(y)
∫ 1/g(y) dy = ∫ f(x) dx
Separate variables
Apply initial condition
Example
dy/dx = 2xy, y(0) = 3
y = 3e^(x²)
Topic 3 · Geometry & Trigonometry
Lines & Planes in 3D
Line: r = a + tb
Plane: n · (r − a) = 0
ax + by + cz = d
Sub t-param into plane
Normal vector n
Example
r = (1,2,3)+t(2,−1,1) meets 3x+y−2z=5 at t=2
Intersection: (5, 0, 5)
Topic 5 · Calculus
Optimisation
Set f'(x) = 0 → solve
f''(x) > 0 → minimum
f''(x) < 0 → maximum
Set up constraint
Check endpoints
Example
Fence 3 sides: 2y+x=200, max A=xy
x=100, y=50, A_max = 5000 m²
Topic 4 · Statistics & Probability
Expected Value & Variance
E(X) = Σ x · P(X=x)
Var(X) = E(X²) − [E(X)]²
SD(X) = √Var(X)
Probability distribution table
Σ P = 1
Example
X: {0,1,2,3}, P: {0.1,0.3,0.4,0.2}
E(X) = 1.7, SD(X) ≈ 0.900
Topic 4 · Statistics & Probability
Hypothesis Testing (t-test)
t = (x̄ − μ₀) / (s/√n)
df = n − 1
p-value < α → reject H₀
Two-tailed: α/2 each side
H₀: μ = μ₀
Example
n=15, x̄=72, s=8, test μ=75 (5%)
t ≈ −1.45, p ≈ 0.168 → fail to reject H₀