Formula | 数学公式

📚 Formula | 数学公式

In A‑Level Mathematics, a formula is not just a string of symbols; it is a precise relationship between quantities, a tool that transforms problems into answers. The Edexcel specification expects you to recall, understand, and apply a core set of formulas – some provided in the formula book, many committed to memory. This article brings together the essential formulas from pure mathematics, organised by topic, with clear explanations and practical tips for using them under exam conditions. Whether you are tackling quadratics, sequences, trigonometry, or calculus, mastering these formulas is the foundation of efficient and accurate problem solving.

在 A‑Level 数学中，公式不仅是一串符号，更是量之间精确关系的表达，是将问题转化为答案的工具。Edexcel 考试大纲要求你记住、理解并应用一组核心公式——有些在公式手册中提供，许多则需要熟记于心。本文汇集了纯数学中的关键公式，按主题整理，并附有清晰的解释和考试实用技巧。无论你面对的是二次方程、数列、三角函数还是微积分，掌握这些公式都是高效准确解题的基础。

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1. Quadratic Formula | 二次公式

on of the form ax² + bx + c = 0 (a ≠ 0), the solutions are given by the quadratic formula:

对于任何形如 ax² + bx + c = 0（a ≠ 0）的二次方程，其解由二次公式给出：

x = [-b ± √b² - 4ac] / (2a)

The expression under the square root sign, b² - 4ac, is called the discriminant. It determines the nature of the roots: if it is positive, there are two distinct real roots; if zero, one repeated real root; if negative, no real roots but two complex conjugate roots. In coordinate geometry, the quadratic formula also gives the x‑coordinates of the intersection points of a parabola with the x‑axis. Learning to substitute coefficients accurately – including negative values – is a fundamental skill. Remember to divide the entire numerator by 2a, not just part of it.

根号下的表达式 b² - 4ac 称为判别式，它决定根的性质：若为正，则有两个不等的实根；若为零，则有一个重根；若为负，则无实根但有两个共轭复根。在坐标几何中，二次公式也给出了抛物线与 x 轴交点的横坐标。准确代入系数（包括负值）是关键技能。记住整个分子要除以 2a，而不仅仅是其中一部分。

Completing the square is an alternative method that leads to the formula: rewrite ax² + bx + c as a(x + b/(2a))² + (c - b²/(4a)). The turning point (vertex) of the parabola is at (-b/(2a), c - b²/(4a)). The quadratic formula then emerges by setting the squared term equal to (b² - 4ac)/(4a²).

配方法是另一种推导公式的方法：将 ax² + bx + c 写成 a(x + b/(2a))² + (c - b²/(4a))。抛物线的顶点坐标为 (-b/(2a), c - b²/(4a))。将平方项设为 (b² - 4ac)/(4a²)，即可得到二次公式。

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2. Arithmetic Sequences | 等差数列

An arithmetic sequence has a constant difference d between consecutive terms. If the first term is a, the n‑th term is:

等差数列的相邻项之差 d 为常数。若首项为 a，则第 n 项为：

uₙ = a + (n - 1)d

The sum of the first n terms, denoted Sₙ, can be found by pairing terms from the ends:

前 n 项之和 Sₙ 可通过首尾配对求得：

Sₙ = n/2 [2a + (n - 1)d] = n/2 (a + l)

is formula is particularly useful when you know the first and last terms but not the common difference. In modelling problems, arithmetic progressions often represent linear growth or decline – orm increase in production. Always check whether a problem describes an arithmetic pattern by verifying that the difference between successive data points is constant.

其中 l = a + (n - 1)d 为末项。当已知首项和末项但不知公差时，此公式尤为有用。在建模问题中，等差数列常表示线性增长或下降，例如单利或产量的均匀增加。务必通过检验相邻数据点之差是否为常数来判断问题是否为等差模型。

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3. Geometric Sequences | 等比数列

A geometric sequence has a constant ratio r between consecutive terms. With first term a, the n‑th term is:

等比数列的相邻项之比 r 为常数。首项为 a 时，第 n 项为：

uₙ = arⁿ⁻¹

For a geometric series, the sum of the first n terms (r ≠ 1) is:

对于等比级数，前 n 项之和 (r ≠ 1) 为：

Sₙ = a(1 - rⁿ)/(1 - r)

If |r| < 1, the infinite geometric series converges, and its sum to infinity is:

若 |r| < 1，则无穷等比级数收敛，其无穷和为：

S∞ = a/(1 - r)

Geometric sequences model exponential growth and decay – for instance, compound interest, population growth, and radioactive decay. In Edexcel exams, you may need to use logarithms to solve for n when given Sₙ, especially when r > 1. Pay careful attention to the case r < 0: the terms alternate in sign, and the convergence condition still requires |r| < 1.

等比数列可模拟指数增长或衰减，例如复利、人口增长和放射性衰变。在 Edexcel 考试中，已知 Sₙ 求解 n 时可能需要使用对数，特别是当 r > 1 时。注意 r < 0 的情况：项会正负交替，而收敛条件仍要求 |r| < 1。

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4. Binomial Expansion | 二项式展开

For any positive integer n, the binomial expansion gives:

对于任意正整数 n，二项式展开为：

(a + b)ⁿ = Σ [from r=0 to n] C(n,r) aⁿ⁻ʳ bʳ

where the binomial coefficient C(n,r) = n! / [r!(n - r)! the number of ways to choose r items from n. You can also obtain coefficients from Pascal' ating expressions like (1 + x)ⁿ when x is small; the first few terms give a good estimate.

其中二项式系数 C(n,r) = n! / [r!(n - r)!] 表示从 n 个中选 r 个的组合数。系数也可由帕斯卡三角形或计算器上的 nCr 键求得。当 x 很小时，此展开式对于近似形如 (1 + x)ⁿ 的表达式尤其有用；前几项即可给出良好估计。

When n is not a positive integer, the expansion becomes an infinite series valid for |x| < 1:

(1 + x)ⁿ = 1 + nx + n(n-1)/2! x² + n(n-1)(n-2)/3! x³ + …

This is particularly powerful in A‑Level for expanding rational powers and for finding approximations. You must state the range of validity whenever you use an infinite binomial expansion; typically |bx/a| < 1 for expressions of the form (a + bx)ⁿ.

当 n 非正整数时，展开成为无穷级数，在 |x| < 1 时成立。这在 A‑Level 中对于展开有理数次幂和求近似值非常有用。使用无穷二项展开时必须注明成立范围；对于形如 (a + bx)ⁿ 的表达式，通常是 |bx/a| < 1。

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5. Trigonometric Identities | 三角恒等式

Trigonometric identities are essential for simplifying expressions and solving equations. The fundamental Pythagorean identity:

三角恒等式对于化简表达式和解方程至关重要。基本勾股恒等式为：

sin²θ + cos²θ ≡ 1

From this, you can derive 1 + tan²θ ≡ sec²θ and 1 + cot²θ ≡ cosec²θ. The tangent identity is:

tanθ ≡ sinθ / cosθ

For compound angles, the addition formulas are heavily tested:

对于和差角，加法公式是考查重点：

sin(A ± B) = sinA cosB ± cosA sinB

cos(A ± B) = cosA cosB ∓ sinA sinB

tan(A ± B) = (tanA ± tanB) / (1 ∓ tanA tanB)

The double‑angle formulas follow directly:

倍角公式可直接推出：

sin2θ = 2sinθ cosθ

cos2θ = cos²θ - sin²θ = 2cos²θ - 1 = 1 - 2sin²θ

as sums, which is vital for integration (e.g., sinA cosB = ½[sin(A+B) + sin(A-B)]). The R‑ (harmonic form) is also crucial: a sinθ ± b cosθ can be written as R sin(θ ± α) or R cos(θ ∓ α), where R = √a² + b² and α = arctan(b/a) adjusted for quadrant.

这些公式能将积化为和差，对积分至关重要（如 sinA cosB = ½[sin(A+B) + sin(A-B)]）。R 公式（辅助角形式）同样关键：a sinθ ± b cosθ 可写成 R sin(θ ± α) 或 R cos(θ ∓ α)，其中 R = √a² + b²，α = arctan(b/a) 并需调整象限。

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6. Differentiation Rules | 微分法则

Differentiation gives the gradient of a curve. For powers of x, the basic rule is:

微分给出曲线的斜率。对于 x 的幂函数，基本法则为：

d/dx (xⁿ) = nxⁿ⁻¹

ative of a sum is the sum of derivatives, and for a constant multiple: d/dx [c f(x)] = c f'(x). For composite functions, the chain rule is essential:

这适用于任何实指数 n。和的导数为导数之和，常数倍法则：d/dx [c f(x)] = c f'(x)。对于复合函数，链式法则至关重要：

dy/dx = dy/du × du/dx

The product rule and quotient rule are used for products and ratios of functions:

乘法法则和商法则用于函数的积和商：

d/dx (uv) = u'v + uv'

d/dx (u/v) = (u'v - uv') / v²

off by heart: d/dx (sin x) = cos x, d/dx (cos x) = -sin x, d/dx (tan x) = sec² x, d/dx (eˣ) = eˣ ot be isolated; differentiate both sides with respect to x, treating y as a function of x and multiplying by dy/dx whenever differentiating a y‑term.

必须熟记标准导数：d/dx (sin x) = cos x, d/dx (cos x) = -sin x, d/dx (tan x) = sec² x, d/dx (eˣ) = eˣ, d/dx (ln x) = 1/x。对于参数方程 x = f(t), y = g(t)，斜率为 dy/dx = (dy/dt) / (dx/dt)。隐函数微分用于无法分离 y 的情形；对 x 两边求导，将 y 视为 x 的函数，每对 y 项求导时乘以 dy/dx。

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7. Integration Essentials | 积分基础公式

Integration is the reverse of differentiation. The power rule for integration (n ≠ -1) is:

积分是微分的逆运算。幂函数积分法则 (n ≠ -1) 为：

∫ xⁿ dx = xⁿ⁺¹/(n+1) + C

where C is the constant of integration. For the special case n = -1:

∫ (1/x) dx = ln |x| + C

derivatives: ∫ cos x dx = sin x + C, ∫ sin x dx = -cos x + C, ∫ sec² x dx = tan x + C, ∫ eˣ dx = eˣ . The definite integral ∫ₐᵇ f(x) dx represents the signed area between the curve and the x‑axis from x = a to x = b. The fundamental theorem of calculus states that if F'(x) = f(x), then ∫ₐᵇ f(x) dx = F(b) - F(a).

标准积分与导数直接对应：∫ cos x dx = sin x + C, ∫ sin x dx = -cos x + C, ∫ sec² x dx = tan x + C, ∫ eˣ dx = eˣ + C。定积分 ∫ₐᵇ f(x) dx 表示曲线与 x 轴之间从 x=a 到 x=b 的有向面积。微积分基本定理指出，若 F'(x) = f(x)，则 ∫ₐᵇ f(x) dx = F(b) - F(a)。

n and integration by parts are the two principal techniques. Substitution: for ∫ f(g(x)) g'(x) dx, let u = g(x) so du = g'(x) dx. Parts formula comes from the product rule:

∫ u dv = uv - ∫ v du

tion is impossible: ∫ₐᵇ y dx ≈ h/2 [y₀ + 2(y₁ + y₂ + ... + yₙ₋₁) + yₙ], where h = (b-a)/n.

换元积分和分部积分是两大主要技巧。换元：对于 ∫ f(g(x)) g'(x) dx，令 u = g(x) 得 du = g'(x) dx。分部积分公式来自乘法法则：∫ u dv = uv - ∫ v du。对于有理代数式，可用部分分式将有理函数拆分为简单分式之和再积分。当无法精确积分时，梯形法则提供数值近似：∫ₐᵇ y dx ≈ h/2 [y₀ + 2(y₁ + y₂ + ... + yₙ₋₁) + yₙ]，其中 h = (b-a)/n。

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8. Exponentials and Logarithms | 指数与对数公式

Exponential functions and logarithms are inverses. The key relationship is:

指数函数与对数互为反函数。关键关系为：

y = aˣ ⇔ x = logₐ y

In A‑Level, we primarily use the natural exponential eˣ eir derivatives and integrals have already been stated. The laws of logarithms are vital for manipulating equations:

在 A‑Level 中，我们主要使用自然指数 eˣ 和自然对数 ln x = logₑ x。其导数和积分已陈述。对数运算律对于方程变形至关重要：

logₐ (xy) = logₐ x + logₐ y

logₐ (x/y) = logₐ x - logₐ y

logₐ (xᵏ) = k logₐ x

ential growth and decay, the model is x = x₀ eᵏᵗ, where k > 0 for growth, k < 0 for decay.

换底公式十分便捷：logₐ b = (log_c b) / (log_c a)。指数增长和衰减的模型为 x = x₀ eᵏᵗ，其中 k > 0 表示增长，k < 0 表示衰减。

When solving equations like aˣ = b, take logs: x ln a = ln b ⇒ x = ln b / ln a. Remember that ln e = 1, and that e^{ln x} = x for x > 0. The exponential form is also used to derive the derivative of aˣ: d/dx (aˣ) = aˣ ln a.

解 aˣ = b 这类方程时两边取对数：x ln a = ln b ⇒ x = ln b / ln a。记住 ln e = 1，且对于 x > 0 有 e^{ln x} = x。指数形式也可用于推导 aˣ 的导数：d/dx (aˣ) = aˣ ln a。

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9. Coordinate Geometry | 坐标几何公式

Coordinate geometry links algebra to geometric shapes. The distance between two points (x₁, y₁) and (x₂, y₂) is given by Pythagoras' theorem:

坐标几何将代数与几何图形联系起来。两点 (x₁, y₁) 和 (x₂, y₂) 之间的距离由勾股定理给出：

d = √[(x₂ - x₁)² + (y₂ - y₁)²]

The midpoint of the segment joining these points is:

连接这两点的线段中点为：

M = ((x₁ + x₂)/2, (y₁ + y₂)/2)

+ by + c = 0. Parallel lines have equal gradients; perpendicular lines satisfy m₁ m₂ = -1.

直线的斜率 m = (y₂ - y₁) / (x₂ - x₁)。直线方程可用多种形式表示：y = mx + c、y - y₁ = m(x - x₁) 或 ax + by + c = 0。平行线斜率相等；垂直线满足 m₁ m₂ = -1。

y - b)² = r². The expanded form x² + y² + 2gx + 2fy + c = 0 has centre (-g, -f) and radius √g² + f² - c. The perpendicular bisector of a chord passes through the centre – a useful property in problem solving. To find intersections between a line and a circle, substitute the line equation into the circle to obtain a quadratic in one variable, then use the discriminant to determine the number of intersection points.

对于圆心在 (a, b)、半径为 r 的圆，方程为 (x - a)² + (y - b)² = r²。展开式 x² + y² + 2gx + 2fy + c = 0 的圆心为 (-g, -f)，半径为 √g² + f² - c。弦的垂直平分线经过圆心——这是解题中的有用性质。求直线与圆的交点时，将直线方程代入圆的方程得到一元二次方程，再用判别式确定交点个数。

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10. Series Summation | 级数求和公式

Beyond arithmetic and geometric series, the Edexcel A‑Level course requires standard summations of powers of integers. These appear in questions on proof by induction and in evaluating Riemann sums:

除等差和等比级数外，Edexcel A‑Level 课程还要求掌握正整数的幂次标准求和公式。这些公式出现在归纳证明和黎曼和求值问题中：

∑ [from r=1 to n] r = n(n+1)/2

∑ [from r=1 to n] r² = n(n+1)(2n+1)/6

∑ [from r=1 to n] r³ = [n(n+1)/2]²

lapses to f(1) - f(n+1). This technique simplifies many summation problems, especially those involving rational functions with denominators that differ by a constant.

这些公式可与线性性质结合，求 ∑ (3r² + 2r - 1) 之类和式。在微积分中，它们构成了定积分作为和式极限的基础。对于差分法，你常需认出形如 ∑ [f(r) - f(r+1)] 的级数可简化为 f(1) - f(n+1)。此技巧可简化许多求和问题，尤其是涉及分母相差常数的有理函数时。

g polynomial approximations to functions near zero. The general formula is f(x) = f(0) + f'(0)x + f''(0)/2! x² + f'''(0)/3! x³ + … . Standard expansions include eˣ = 1 + x + x²/2! + x³/3! + … , sin x = x - x³/3! + x⁵/5! - … , cos x = 1 - x²/2! + x⁴/4! - … , and ln(1+x) = x - x²/2 + x³/3 - … (|x| < 1). These are valid for all real x in the case of eˣ, sin, cos, but only for |x| < 1 for ln(1+x).

麦克劳林级数是另一类重要的无穷级数，给出函数在零点附近的多项式逼近。一般公式为 f(x) = f(0) + f'(0)x + f''(0)/2! x² + f'''(0)/3! x³ + … 。标准展开包括 eˣ = 1 + x + x²/2! + x³/3! + …、sin x = x - x³/3! + x⁵/5! - …、cos x = 1 - x²/2! + x⁴/4! - … 以及 ln(1+x) = x - x²/2 + x³/3 - … (|x| < 1)。其中 eˣ、sin、cos 对所有实数 x 成立，而 ln(1+x) 仅对 |x| < 1 成立。

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