CCEA Pre-U Further Mathematics: Formula & Theorem Quick Reference Handbook – CCEA Pre-U 进阶数学：公式定理速查手册
CCEA Pre-U 进阶数学（Further Mathematics）是一门极具挑战性的课程，涵盖纯数学、力学和统计学三大领域。本文整理了考试中最常用的核心公式与定理，按模块分类，便于考前快速查阅与记忆。
CCEA Pre-U Further Mathematics is a highly demanding qualification, covering Pure Mathematics, Mechanics, and Statistics. This article compiles the most essential formulas and theorems across all modules, organized by topic for quick revision and memorization before the exam.
一、代数与函数 / Algebra and Functions
二次方程求根公式 / Quadratic Formula
对于 ax² + bx + c = 0 (a ≠ 0)：x = [-b ± √(b² – 4ac)] / 2a
For ax² + bx + c = 0 (a ≠ 0): x = [-b ± √(b² – 4ac)] / 2a
判别式 / Discriminant
Δ = b² – 4ac。Δ > 0：两个不等实根；Δ = 0：一个重根；Δ < 0：无实根（两个共轭复根）。
Δ = b² – 4ac. Δ > 0: two distinct real roots; Δ = 0: one repeated root; Δ < 0: no real roots (two complex conjugate roots).
多项式余数定理 / Polynomial Remainder Theorem
多项式 f(x) 除以 (x – a) 的余数为 f(a)。
When a polynomial f(x) is divided by (x – a), the remainder is f(a).
因式定理 / Factor Theorem
若 f(a) = 0，则 (x – a) 是 f(x) 的一个因式。
If f(a) = 0, then (x – a) is a factor of f(x).
二项式展开 / Binomial Expansion
(1 + x)^n = 1 + nx + n(n-1)x²/2! + n(n-1)(n-2)x³/3! + … (|x| < 1，n 为任意实数)。
(1 + x)^n = 1 + nx + n(n-1)x²/2! + n(n-1)(n-2)x³/3! + … (|x| < 1, n is any real number).
部分分式分解 / Partial Fractions
线性因子：1/[(x+a)(x+b)] ≡ A/(x+a) + B/(x+b)；重复因子：f(x)/(x+a)² ≡ A/(x+a) + B/(x+a)²。
Linear factors: 1/[(x+a)(x+b)] ≡ A/(x+a) + B/(x+b); Repeated factors: f(x)/(x+a)² ≡ A/(x+a) + B/(x+a)².
指数与对数法则 / Laws of Exponents and Logarithms
a^x · a^y = a^(x+y)；a^x / a^y = a^(x-y)；(a^x)^y = a^(xy)；log_a(xy) = log_a(x) + log_a(y)；log_a(x/y) = log_a(x) – log_a(y)；log_a(x^n) = n·log_a(x)；换底公式：log_a(b) = log_c(b) / log_c(a)。
a^x · a^y = a^(x+y); a^x / a^y = a^(x-y); (a^x)^y = a^(xy); log_a(xy) = log_a(x) + log_a(y); log_a(x/y) = log_a(x) – log_a(y); log_a(x^n) = n·log_a(x); Change of base: log_a(b) = log_c(b) / log_c(a).
二、三角学 / Trigonometry
基本恒等式 / Fundamental Identities
sin²θ + cos²θ = 1；tanθ = sinθ / cosθ；1 + tan²θ = sec²θ；1 + cot²θ = cosec²θ。
sin²θ + cos²θ = 1; tanθ = sinθ / cosθ; 1 + tan²θ = sec²θ; 1 + cot²θ = cosec²θ.
和角公式 / Compound Angle Formulae
sin(A ± B) = sinA cosB ± cosA sinB；cos(A ± B) = cosA cosB ∓ sinA sinB；tan(A ± B) = (tanA ± tanB) / (1 ∓ tanA tanB)。
sin(A ± B) = sinA cosB ± cosA sinB; cos(A ± B) = cosA cosB ∓ sinA sinB; tan(A ± B) = (tanA ± tanB) / (1 ∓ tanA tanB).
倍角公式 / Double Angle Formulae
sin2A = 2 sinA cosA；cos2A = cos²A – sin²A = 2cos²A – 1 = 1 – 2sin²A；tan2A = 2tanA / (1 – tan²A)。
sin2A = 2 sinA cosA; cos2A = cos²A – sin²A = 2cos²A – 1 = 1 – 2sin²A; tan2A = 2tanA / (1 – tan²A).
和差化积 / Sum-to-Product Formulae
sinP + sinQ = 2 sin[(P+Q)/2] cos[(P-Q)/2]；sinP – sinQ = 2 cos[(P+Q)/2] sin[(P-Q)/2]；cosP + cosQ = 2 cos[(P+Q)/2] cos[(P-Q)/2]；cosP – cosQ = -2 sin[(P+Q)/2] sin[(P-Q)/2]。
sinP + sinQ = 2 sin[(P+Q)/2] cos[(P-Q)/2]; sinP – sinQ = 2 cos[(P+Q)/2] sin[(P-Q)/2]; cosP + cosQ = 2 cos[(P+Q)/2] cos[(P-Q)/2]; cosP – cosQ = -2 sin[(P+Q)/2] sin[(P-Q)/2].
正弦定理与余弦定理 / Sine Rule and Cosine Rule
正弦定理：a/sinA = b/sinB = c/sinC = 2R。余弦定理：a² = b² + c² – 2bc cosA。
Sine Rule: a/sinA = b/sinB = c/sinC = 2R. Cosine Rule: a² = b² + c² – 2bc cosA.
弧度制 / Radian Measure
180° = π rad；弧长 s = rθ；扇形面积 A = ½ r²θ。
180° = π rad; Arc length s = rθ; Sector area A = ½ r²θ.
三、微积分 / Calculus
基本求导公式 / Standard Derivatives
d/dx (x^n) = nx^(n-1)；d/dx (e^x) = e^x；d/dx (ln x) = 1/x；d/dx (sin x) = cos x；d/dx (cos x) = -sin x；d/dx (tan x) = sec²x；d/dx (sec x) = sec x tan x；d/dx (cosec x) = -cosec x cot x；d/dx (cot x) = -cosec²x；d/dx (a^x) = a^x ln a；d/dx (arcsin x) = 1/√(1-x²)；d/dx (arccos x) = -1/√(1-x²)；d/dx (arctan x) = 1/(1+x²)。
d/dx (x^n) = nx^(n-1); d/dx (e^x) = e^x; d/dx (ln x) = 1/x; d/dx (sin x) = cos x; d/dx (cos x) = -sin x; d/dx (tan x) = sec²x; d/dx (sec x) = sec x tan x; d/dx (cosec x) = -cosec x cot x; d/dx (cot x) = -cosec²x; d/dx (a^x) = a^x ln a; d/dx (arcsin x) = 1/√(1-x²); d/dx (arccos x) = -1/√(1-x²); d/dx (arctan x) = 1/(1+x²).
乘法法则 / Product Rule
d/dx (uv) = u(dv/dx) + v(du/dx)。
d/dx (uv) = u(dv/dx) + v(du/dx).
除法法则 / Quotient Rule
d/dx (u/v) = [v(du/dx) – u(dv/dx)] / v²。
d/dx (u/v) = [v(du/dx) – u(dv/dx)] / v².
链式法则 / Chain Rule
dy/dx = dy/du · du/dx。
dy/dx = dy/du · du/dx.
基本积分公式 / Standard Integrals
∫ x^n dx = x^(n+1)/(n+1) + C (n ≠ -1)；∫ 1/x dx = ln|x| + C；∫ e^x dx = e^x + C；∫ a^x dx = a^x/ln a + C；∫ sin x dx = -cos x + C；∫ cos x dx = sin x + C；∫ sec²x dx = tan x + C；∫ cosec²x dx = -cot x + C；∫ sec x tan x dx = sec x + C；∫ cosec x cot x dx = -cosec x + C。
∫ x^n dx = x^(n+1)/(n+1) + C (n ≠ -1); ∫ 1/x dx = ln|x| + C; ∫ e^x dx = e^x + C; ∫ a^x dx = a^x/ln a + C; ∫ sin x dx = -cos x + C; ∫ cos x dx = sin x + C; ∫ sec²x dx = tan x + C; ∫ cosec²x dx = -cot x + C; ∫ sec x tan x dx = sec x + C; ∫ cosec x cot x dx = -cosec x + C.
分部积分 / Integration by Parts
∫ u(dv/dx) dx = uv – ∫ v(du/dx) dx 或简写为 ∫ u dv = uv – ∫ v du。
∫ u(dv/dx) dx = uv – ∫ v(du/dx) dx, or in shorthand: ∫ u dv = uv – ∫ v du.
换元积分 / Integration by Substitution
∫ f(g(x)) g'(x) dx = ∫ f(u) du，其中 u = g(x)。
∫ f(g(x)) g'(x) dx = ∫ f(u) du, where u = g(x).
参数方程求导 / Parametric Differentiation
若 x = f(t), y = g(t)，则 dy/dx = (dy/dt) / (dx/dt) = g'(t)/f'(t)。
If x = f(t), y = g(t), then dy/dx = (dy/dt) / (dx/dt) = g'(t)/f'(t).
隐函数求导 / Implicit Differentiation
对等式两边同时对 x 求导，将 dy/dx 项集中到一侧求解。
Differentiate both sides with respect to x, collect dy/dx terms on one side and solve.
定积分求面积与体积 / Definite Integrals: Area and Volume
曲线下方面积：A = ∫[a→b] y dx。绕 x 轴旋转体体积：V = π ∫[a→b] y² dx。绕 y 轴旋转体体积：V = π ∫[c→d] x² dy。
Area under a curve: A = ∫[a→b] y dx. Volume of revolution about x-axis: V = π ∫[a→b] y² dx. Volume of revolution about y-axis: V = π ∫[c→d] x² dy.
牛顿-拉弗森迭代法 / Newton-Raphson Method
x_{n+1} = x_n – f(x_n)/f'(x_n)。
x_{n+1} = x_n – f(x_n)/f'(x_n).
四、向量与矩阵 / Vectors and Matrices
向量基本运算 / Basic Vector Operations
点积（内积）：a · b = |a||b|cosθ = a₁b₁ + a₂b₂ + a₃b₃。叉积（外积）：a × b = |a||b|sinθ n̂（方向由右手定则确定）。|a × b| = 以 a,b 为边的平行四边形面积。
Dot product: a · b = |a||b|cosθ = a₁b₁ + a₂b₂ + a₃b₃. Cross product: a × b = |a||b|sinθ n̂ (direction given by right-hand rule). |a × b| = area of parallelogram with sides a, b.
向量方程 / Vector Equations
直线方程：r = a + λb（a 为线上一点，b 为方向向量）。平面方程：r · n = d（n 为法向量）。
Line equation: r = a + λb (a is a point on the line, b is the direction vector). Plane equation: r · n = d (n is the normal vector).
2×2 矩阵行列式 / Determinant of a 2×2 Matrix
det[[a, b], [c, d]] = ad – bc。
det[[a, b], [c, d]] = ad – bc.
2×2 逆矩阵 / Inverse of a 2×2 Matrix
若 M = [[a, b], [c, d]] 且 det(M) ≠ 0，则 M⁻¹ = 1/(ad-bc) [[d, -b], [-c, a]]。
If M = [[a, b], [c, d]] and det(M) ≠ 0, then M⁻¹ = 1/(ad-bc) [[d, -b], [-c, a]].
3×3 矩阵行列式 / Determinant of a 3×3 Matrix
det[[a,b,c],[d,e,f],[g,h,i]] = a(ei-fh) – b(di-fg) + c(dh-eg)。
det[[a,b,c],[d,e,f],[g,h,i]] = a(ei-fh) – b(di-fg) + c(dh-eg).
矩阵变换 / Matrix Transformations
旋转（逆时针 θ）：[[cosθ, -sinθ], [sinθ, cosθ]]；反射（关于 y=x）：[[0,1],[1,0]]；缩放（因子 k）：[[k,0],[0,k]]。
Rotation (anticlockwise θ): [[cosθ, -sinθ], [sinθ, cosθ]]; Reflection (in y=x): [[0,1],[1,0]]; Scaling (factor k): [[k,0],[0,k]].
特征值与特征向量 / Eigenvalues and Eigenvectors
对于矩阵 A，若 Av = λv（v ≠ 0），则 λ 为特征值，v 为对应的特征向量。特征方程：det(A – λI) = 0。
For a matrix A, if Av = λv (v ≠ 0), then λ is an eigenvalue and v is the corresponding eigenvector. Characteristic equation: det(A – λI) = 0.
五、复数 / Complex Numbers
基本形式 / Basic Form
z = a + bi，其中 i² = -1。实部 Re(z) = a，虚部 Im(z) = b。共轭复数 z* = a – bi。模 |z| = √(a² + b²)。幅角 arg(z) = arctan(b/a)，注意象限。
z = a + bi, where i² = -1. Real part Re(z) = a, imaginary part Im(z) = b. Complex conjugate z* = a – bi. Modulus |z| = √(a² + b²). Argument arg(z) = arctan(b/a), taking account of the quadrant.
极坐标形式 / Polar Form
z = r(cosθ + i sinθ) = r·cis(θ)，其中 r = |z|，θ = arg(z)。
z = r(cosθ + i sinθ) = r·cis(θ), where r = |z|, θ = arg(z).
欧拉公式 / Euler’s Formula
e^(iθ) = cosθ + i sinθ。因此 z = re^(iθ)。
e^(iθ) = cosθ + i sinθ. Hence z = re^(iθ).
棣莫弗定理 / De Moivre’s Theorem
[r(cosθ + i sinθ)]^n = r^n [cos(nθ) + i sin(nθ)]。特别地，(cosθ + i sinθ)^n = cos(nθ) + i sin(nθ)。
[r(cosθ + i sinθ)]^n = r^n [cos(nθ) + i sin(nθ)]. In particular, (cosθ + i sinθ)^n = cos(nθ) + i sin(nθ).
单位根 / Roots of Unity
方程 z^n = 1 的 n 个根为 z_k = e^(2πk i / n)，k = 0, 1, …, n-1。所有根均匀分布在复平面单位圆上。
The n roots of z^n = 1 are z_k = e^(2πk i / n), k = 0, 1, …, n-1. All roots are evenly spaced on the unit circle in the complex plane.
六、微分方程 / Differential Equations
一阶线性微分方程 / First-Order Linear Differential Equations
dy/dx + P(x)y = Q(x)。积分因子 IF = e^(∫P(x)dx)。通解：y·IF = ∫ Q(x)·IF dx + C。
dy/dx + P(x)y = Q(x). Integrating factor IF = e^(∫P(x)dx). General solution: y·IF = ∫ Q(x)·IF dx + C.
可分离变量微分方程 / Separable Differential Equations
形式：dy/dx = f(x)g(y) 即 (1/g(y)) dy = f(x) dx。两边积分求解。
Form: dy/dx = f(x)g(y), i.e. (1/g(y)) dy = f(x) dx. Integrate both sides to solve.
二阶线性常系数齐次微分方程 / Second-Order Linear Homogeneous DE with Constant Coefficients
a(d²y/dx²) + b(dy/dx) + cy = 0。特征方程：am² + bm + c = 0。
a(d²y/dx²) + b(dy/dx) + cy = 0. Auxiliary equation: am² + bm + c = 0.
若判别式 > 0（两不等实根 m₁, m₂）：y = Ae^(m₁x) + Be^(m₂x)。
If discriminant > 0 (two distinct real roots m₁, m₂): y = Ae^(m₁x) + Be^(m₂x).
若判别式 = 0（重根 m）：y = (A + Bx)e^(mx)。
If discriminant = 0 (repeated root m): y = (A + Bx)e^(mx).
若判别式 < 0（共轭复根 α ± βi）：y = e^(αx)(A cos βx + B sin βx)。
If discriminant < 0 (complex conjugate roots α ± βi): y = e^(αx)(A cos βx + B sin βx).
七、极坐标 / Polar Coordinates
极坐标与笛卡尔坐标转换 / Polar to Cartesian Conversion
x = r cosθ，y = r sinθ；r = √(x² + y²)，θ = arctan(y/x)。
x = r cosθ, y = r sinθ; r = √(x² + y²), θ = arctan(y/x).
极坐标曲线面积 / Area in Polar Coordinates
A = ½ ∫[α→β] r² dθ。
A = ½ ∫[α→β] r² dθ.
八、双曲函数 / Hyperbolic Functions
定义 / Definitions
sinh x = (e^x – e^(-x)) / 2；cosh x = (e^x + e^(-x)) / 2；tanh x = sinh x / cosh x = (e^x – e^(-x)) / (e^x + e^(-x))。
sinh x = (e^x – e^(-x)) / 2; cosh x = (e^x + e^(-x)) / 2; tanh x = sinh x / cosh x = (e^x – e^(-x)) / (e^x + e^(-x)).
基本恒等式 / Fundamental Identity
cosh²x – sinh²x = 1。
cosh²x – sinh²x = 1.
导数 / Derivatives
d/dx (sinh x) = cosh x；d/dx (cosh x) = sinh x；d/dx (tanh x) = sech²x。
d/dx (sinh x) = cosh x; d/dx (cosh x) = sinh x; d/dx (tanh x) = sech²x.
反双曲函数 / Inverse Hyperbolic Functions
arsinh x = ln(x + √(x² + 1))；arcosh x = ln(x + √(x² – 1)), x ≥ 1；artanh x = ½ ln[(1+x)/(1-x)], |x| < 1。
arsinh x = ln(x + √(x² + 1)); arcosh x = ln(x + √(x² – 1)), x ≥ 1; artanh x = ½ ln[(1+x)/(1-x)], |x| < 1.
九、力学 / Mechanics
运动学（匀加速直线运动）/ Kinematics (Constant Acceleration – SUVAT)
v = u + at；s = ut + ½ at²；s = ½ (u + v)t；v² = u² + 2as；s = vt – ½ at²。
v = u + at; s = ut + ½ at²; s = ½ (u + v)t; v² = u² + 2as; s = vt – ½ at².
其中 u = 初速度，v = 末速度，a = 加速度，s = 位移，t = 时间。
Where u = initial velocity, v = final velocity, a = acceleration, s = displacement, t = time.
微积分形式 / Calculus Form (Variable Acceleration)
v = ds/dt；a = dv/dt = d²s/dt²；s = ∫ v dt；v = ∫ a dt。
v = ds/dt; a = dv/dt = d²s/dt²; s = ∫ v dt; v = ∫ a dt.
牛顿第二定律 / Newton’s Second Law
F = ma（合外力 = 质量 × 加速度）。
F = ma (resultant force = mass × acceleration).
动量与冲量 / Momentum and Impulse
动量 p = mv。冲量 J = FΔt = Δp = mv – mu。动量守恒：若无外力，Σmv 前 = Σmv 后。
Momentum p = mv. Impulse J = FΔt = Δp = mv – mu. Conservation of momentum: in the absence of external forces, Σmv before = Σmv after.
功与能量 / Work and Energy
功 W = Fs cosθ（θ 为力与位移的夹角）。动能 KE = ½ mv²。重力势能 GPE = mgh。动能定理：W_net = ΔKE。
Work W = Fs cosθ (θ is the angle between force and displacement). Kinetic energy KE = ½ mv². Gravitational potential energy GPE = mgh. Work-Energy Theorem: W_net = ΔKE.
功率 / Power
P = W/t = Fv（F 为驱动力，v 为速度）。
P = W/t = Fv (F is the driving force, v is the velocity).
抛体运动 / Projectile Motion
水平分量：x = u cosθ · t（匀速）；竖直分量：y = u sinθ · t – ½ gt²。飞行时间 T = 2u sinθ / g；最大高度 H = u² sin²θ / (2g)；射程 R = u² sin(2θ) / g。
Horizontal component: x = u cosθ · t (constant velocity); Vertical component: y = u sinθ · t – ½ gt². Time of flight T = 2u sinθ / g; Maximum height H = u² sin²θ / (2g); Range R = u² sin(2θ) / g.
圆周运动 / Circular Motion
角速度 ω = dθ/dt = v/r。向心加速度 a = v²/r = rω²。向心力 F = mv²/r = mrω²。
Angular velocity ω = dθ/dt = v/r. Centripetal acceleration a = v²/r = rω². Centripetal force F = mv²/r = mrω².
简谐运动 / Simple Harmonic Motion (SHM)
加速度 a = -ω²x。位移 x = A sin(ωt) 或 x = A cos(ωt)。速度 v = ±ω√(A² – x²)。最大速度 v_max = ωA。周期 T = 2π/ω。
Acceleration a = -ω²x. Displacement x = A sin(ωt) or x = A cos(ωt). Velocity v = ±ω√(A² – x²). Maximum velocity v_max = ωA. Period T = 2π/ω.
力矩与平衡 / Moments and Equilibrium
力矩 = 力 × 垂直距离。平衡条件：ΣF = 0，ΣM = 0。
Moment = Force × perpendicular distance. Conditions for equilibrium: ΣF = 0, ΣM = 0.
质心 / Centre of Mass
质点系：x̄ = Σ(m_i x_i) / Σm_i。均匀薄板可通过对称性和积分求质心。
System of particles: x̄ = Σ(m_i x_i) / Σm_i. For uniform laminas, the centre of mass can be found through symmetry and integration.
十、统计学 / Statistics
集中趋势度量 / Measures of Central Tendency
平均数（均值）：μ = Σx / n（总体），x̄ = Σx / n（样本）。中位数：排序后中间值。众数：出现频率最高的值。
Mean: μ = Σx / n (population), x̄ = Σx / n (sample). Median: middle value after sorting. Mode: most frequent value.
离散程度度量 / Measures of Dispersion
方差：σ² = Σ(x – μ)² / n（总体），s² = Σ(x – x̄)² / (n-1)（样本）。标准差 = √方差。
Variance: σ² = Σ(x – μ)² / n (population), s² = Σ(x – x̄)² / (n-1) (sample). Standard deviation = √variance.
概率基本公式 / Basic Probability Formulae
P(A∪B) = P(A) + P(B) – P(A∩B)。条件概率：P(A|B) = P(A∩B) / P(B)。独立事件：P(A∩B) = P(A) × P(B)。互斥事件：P(A∩B) = 0。
P(A∪B) = P(A) + P(B) – P(A∩B). Conditional probability: P(A|B) = P(A∩B) / P(B). Independent events: P(A∩B) = P(A) × P(B). Mutually exclusive events: P(A∩B) = 0.
排列与组合 / Permutations and Combinations
排列：ⁿPᵣ = n!/(n-r)!（有序选取）。组合：ⁿCᵣ = C(n,r) = n!/[r!(n-r)!]（无序选取）。
Permutations: ⁿPᵣ = n!/(n-r)! (ordered selection). Combinations: ⁿCᵣ = C(n,r) = n!/[r!(n-r)!] (unordered selection).
二项分布 / Binomial Distribution
X ~ B(n, p)：P(X = r) = C(n,r) p^r (1-p)^(n-r)。E(X) = np；Var(X) = np(1-p)。
X ~ B(n, p): P(X = r) = C(n,r) p^r (1-p)^(n-r). E(X) = np; Var(X) = np(1-p).
泊松分布 / Poisson Distribution
X ~ Po(λ)：P(X = r) = e^(-λ) λ^r / r!。E(X) = λ；Var(X) = λ。
X ~ Po(λ): P(X = r) = e^(-λ) λ^r / r!. E(X) = λ; Var(X) = λ.
正态分布 / Normal Distribution
X ~ N(μ, σ²)。标准化：Z = (X – μ)/σ ~ N(0,1)。约 68% 数据在 μ±σ；约 95% 在 μ±2σ；约 99.7% 在 μ±3σ。
X ~ N(μ, σ²). Standardization: Z = (X – μ)/σ ~ N(0,1). Approximately 68% of data within μ±σ; 95% within μ±2σ; 99.7% within μ±3σ.
连续均匀分布 / Continuous Uniform Distribution
X ~ U(a, b)：pdf f(x) = 1/(b-a), a ≤ x ≤ b。E(X) = (a+b)/2；Var(X) = (b-a)²/12。
X ~ U(a, b): pdf f(x) = 1/(b-a), a ≤ x ≤ b. E(X) = (a+b)/2; Var(X) = (b-a)²/12.
线性组合的期望与方差 / Expectation and Variance of Linear Combinations
E(aX + bY) = aE(X) + bE(Y)。若 X, Y 独立：Var(aX + bY) = a²Var(X) + b²Var(Y)。
E(aX + bY) = aE(X) + bE(Y). If X, Y are independent: Var(aX + bY) = a²Var(X) + b²Var(Y).
十一、级数与序列 / Series and Sequences
等差数列 / Arithmetic Sequences
通项 u_n = a + (n-1)d。前 n 项和 S_n = n/2 [2a + (n-1)d] = n/2 (a + l)。
General term u_n = a + (n-1)d. Sum of first n terms S_n = n/2 [2a + (n-1)d] = n/2 (a + l).
等比数列 / Geometric Sequences
通项 u_n = ar^(n-1)。前 n 项和 S_n = a(1-r^n)/(1-r), r ≠ 1。无穷等比级数（|r| < 1）：S_∞ = a/(1-r)。
General term u_n = ar^(n-1). Sum of first n terms S_n = a(1-r^n)/(1-r), r ≠ 1. Infinite geometric series (|r| < 1): S_∞ = a/(1-r).
麦克劳林级数 / Maclaurin Series
f(x) = f(0) + f'(0)x + f”(0)x²/2! + f”'(0)x³/3! + … + f^(n)(0)x^n/n! + …
f(x) = f(0) + f'(0)x + f”(0)x²/2! + f”'(0)x³/3! + … + f^(n)(0)x^n/n! + …
重要展开：e^x = 1 + x + x²/2! + x³/3! + …；sin x = x – x³/3! + x⁵/5! – …；cos x = 1 – x²/2! + x⁴/4! – …；ln(1+x) = x – x²/2 + x³/3 – … (|x| < 1)。
Important expansions: e^x = 1 + x + x²/2! + x³/3! + …; sin x = x – x³/3! + x⁵/5! – …; cos x = 1 – x²/2! + x⁴/4! – …; ln(1+x) = x – x²/2 + x³/3 – … (|x| < 1).
十二、数值方法 / Numerical Methods
梯形法则 / Trapezium Rule
∫[a→b] f(x) dx ≈ h/2 [y₀ + y_n + 2(y₁ + y₂ + … + y_{n-1})]，其中 h = (b-a)/n。
∫[a→b] f(x) dx ≈ h/2 [y₀ + y_n + 2(y₁ + y₂ + … + y_{n-1})], where h = (b-a)/n.
欧拉方法 / Euler’s Method
对于 dy/dx = f(x,y)，y(x₀) = y₀：y_{n+1} = y_n + h f(x_n, y_n)，其中 h 为步长。
For dy/dx = f(x,y), y(x₀) = y₀: y_{n+1} = y_n + h f(x_n, y_n), where h is the step size.
十三、不等式与线性规划 / Inequalities and Linear Programming
线性规划求解步骤 / Linear Programming Solution Steps
1. 定义决策变量 x 和 y。2. 将约束条件写成不等式。3. 画出可行区域。4. 确定目标函数（最大化或最小化）。5. 在可行域顶点处检验目标函数值。6. 作出结论。
1. Define decision variables x and y. 2. Express constraints as inequalities. 3. Draw the feasible region. 4. Identify the objective function (maximize or minimize). 5. Test the objective function at vertices of the feasible region. 6. State the conclusion.
十四、证明方法 / Methods of Proof
直接证明 / Direct Proof
从已知条件出发，通过逻辑推导得出结论。
Start from known conditions and derive the conclusion through logical steps.
反证法 / Proof by Contradiction
假设命题的否定为真，然后推导出矛盾，从而原命题成立。经典例子：证明 √2 是无理数。
Assume the negation of the proposition is true, then derive a contradiction, thereby proving the original proposition. Classic example: proving √2 is irrational.
归纳法 / Proof by Induction
基例：证明 n=1 时命题成立。归纳步：假设 n=k 时命题成立，证明 n=k+1 时也成立。结论：对所有正整数 n 命题成立。经典例子：证明 Σr = n(n+1)/2。
Base case: prove the statement is true for n=1. Inductive step: assume true for n=k, prove true for n=k+1. Conclusion: the statement is true for all positive integers n. Classic example: proving Σr = n(n+1)/2.
考试技巧 / Exam Tips
公式手册使用策略 / Formula Booklet Strategy
CCEA Pre-U 考试提供公式手册，但手册并不包含所有公式。本文中的双曲函数导数、极坐标面积公式、一阶线性微分方程的积分因子等需要你牢记。考试前务必反复练习，形成肌肉记忆。
The CCEA Pre-U exam provides a formula booklet, but it does not contain all formulas. Hyperbolic function derivatives, polar coordinate area formulas, integrating factors for first-order linear DEs, and several others in this article need to be memorised. Practise repeatedly before the exam to build muscle memory.
常见失分点 / Common Mistakes
1. 忘记积分常数 +C – 不定积分题必扣分。2. 复数幅角忘记考虑象限 – 使用 arctan 后必须画图确认。3. 微分方程特征方程判别式为零时解法特殊，容易误用。4. 抛体运动混淆 sin 和 cos – 水平分量永远是 cos。5. 统计题混淆总体方差和样本方差的分母。
1. Forgetting the constant of integration +C – guaranteed mark loss on indefinite integrals. 2. Neglecting quadrant correction for complex arguments – always sketch after using arctan. 3. The repeated root case in DE auxiliary equations has a special solution form that is easily misapplied. 4. Mixing sin and cos in projectile motion – the horizontal component always uses cos. 5. Confusing the denominators for population variance (n) and sample variance (n-1) in statistics questions.
时间管理 / Time Management
CCEA Pre-U 进阶数学考试时间紧张。建议：纯数学部分（Paper 1 和 Paper 2）每道题约 10-15 分钟；力学/统计部分（Paper 3）每道题约 8-12 分钟。遇到难题不要恋战，先做会做的题，最后回来攻克难点。
Time is tight in the CCEA Pre-U Further Mathematics exam. Recommendation: Pure Mathematics (Papers 1 and 2) allocate approximately 10-15 minutes per question; Mechanics/Statistics (Paper 3) allocate approximately 8-12 minutes. Don’t get stuck on difficult questions – complete the ones you can do first, then return to the challenging ones.
注：本文涵盖了 CCEA Pre-U 进阶数学（Further Mathematics）课程的核心公式与定理。建议打印随身携带，考前反复翻阅。所有公式均基于官方课程大纲核实。
Note: This article covers the core formulas and theorems for the CCEA Pre-U Further Mathematics specification. We recommend printing it and reviewing it frequently before the exam. All formulas have been verified against the official specification.