📚 A-Level Maths: Binomial Expansion of (1+x)^n | A-Level 数学：(1+x)^n 的二项式展开
The binomial expansion is one of the most powerful and frequently tested tools in A-Level mathematics. Its most common form is the expansion of (1+x)^n, and nearly every exam paper asks students to manipulate this expression confidently.
二项式展开是 A-Level 数学中最重要、考查频率最高的工具之一。其中最核心的形式是 (1+x)^n 的展开，几乎每份试卷都要求学生能够自信地处理这一表达式。
Depending on whether n is a non-negative integer or a rational number, the expansion is either finite or infinite. Both cases appear in algebra, calculus, sequences and series, and probability.
根据 n 是非负整数还是有理数，展开式既可能是有限项，也可能是无穷级数。这两种情况都会出现在代数、微积分、数列与级数以及概率统计中。
2. The Expansion for Positive Integer n | 正整数 n 的展开
When n is a non-negative integer, (1+x)^n can be written as a finite sum with exactly n+1 terms:
当 n 是非负整数时，(1+x)^n 可以写成一个恰好包含 n+1 项的有限和：
(1+x)^n = C(n,0) + C(n,1)x + C(n,2)x² + … + C(n,n)xⁿ
Here the symbols C(n,0), C(n,1), C(n,2), … are called binomial coefficients. They can also be written as ⁿCᵣ, nCr, or even displayed in Pascal’s triangle.
这里的 C(n,0)、C(n,1)、C(n,2) 等被称为二项式系数。它们也可以写成 ⁿCᵣ、nCr，或直接显示在杨辉三角中。
For a positive integer n, the expansion terminates at the term xⁿ, and there is no restriction on the value of x. The formula is valid for every real number x.
当 n 为正整数时，展开式在 xⁿ 项终止，并且对 x 的取值没有限制。公式对所有实数 x 都成立。
A particularly useful check is to set x = 1. This gives 2ⁿ = C(n,0) + C(n,1) + C(n,2) + … + C(n,n), showing that the sum of all coefficients is 2ⁿ.
一个特别有用的检验方法是令 x = 1。此时 2ⁿ = C(n,0) + C(n,1) + C(n,2) + … + C(n,n)，说明所有系数之和等于 2ⁿ。
3. Factorials and the nCr Formula | 阶乘与 nCr 公式
The binomial coefficients C(n,r) are defined using factorials. For integers n and r with 0 ≤ r ≤ n:
二项式系数 C(n,r) 利用阶乘来定义。对于满足 0 ≤ r ≤ n 的整数 n 与 r：
C(n,r) = n! / (r!(n-r)!)
The symbol n! means n factorial, which is the product n × (n−1) × (n−2) × … × 3 × 2 × 1. For example, 5! = 5 × 4 × 3 × 2 × 1 = 120.
符号 n! 表示 n 的阶乘，即 n × (n−1) × (n−2) × … × 3 × 2 × 1。例如，5! = 5 × 4 × 3 × 2 × 1 = 120。
By convention, 0! = 1. This convention ensures that C(n,0) = n!/(0! n!) = 1 and C(n,n) = 1, which matches the constant term and the leading term of the expansion.
按照约定，0! = 1。这个约定保证了 C(n,0) = n!/(0! n!) = 1，并且 C(n,n) = 1，这与展开式的常数项和最高次项完全一致。
Two symmetry relations are also essential in exams: C(n,r) = C(n,n−r), and the recurrence relation C(n,r) + C(n,r+1) = C(n+1,r+1) generates Pascal’s triangle.
在考试中还有两个重要的对称性质：C(n,r) = C(n,n−r)，以及递推关系 C(n,r) + C(n,r+1) = C(n+1,r+1)，后者正是杨辉三角的构造依据。
4. The General Term | 一般项
When writing binomial expansions, it is often necessary to find a single term rather than the whole expansion. The general term of the expansion of (1+x)^n is term number r+1:
在书写二项式展开时，我们经常需要寻找某一项，而不是展开全部。对于 (1+x)^n，第 r+1 项为一般项：
Tr+1 = C(n,r)xr
For example, in the expansion of (1+x)^10, the term containing x³ is C(10,3)x³ = 120x³. The coefficient is 120.
例如，在 (1+x)^10 的展开式中，含 x³ 的项是 C(10,3)x³ = 120x³，其系数为 120。
To find the constant term, set r = 0. To find the coefficient of xᵏ, set r = k and evaluate C(n,k). In more advanced questions, you may need to set the power of x equal to a specific index and solve for r.
要求常数项时，令 r = 0；要求 xᵏ 的系数时，令 r = k 并计算 C(n,k)。在进阶题目中，你可能需要令 x 的指数等于某个特定值，然后解出 r。
5. Pascal’s Triangle and Coefficients | 杨辉三角与系数
Pascal’s triangle provides a quick way to list binomial coefficients for small values of n. The rows of the triangle begin:
杨辉三角为较小的 n 值提供了一种快速列出二项式系数的方法。该三角形的前几行如下：
| n = 0 | 1 |
| n = 1 | 1 1 |
| n = 2 | 1 2 1 |
| n = 3 | 1 3 3 1 |
| n = 4 | 1 4 6 4 1 |
| n = 5 | 1 5 10 10 5 1 |
Each row begins and ends with 1, and every interior number is obtained by adding the two numbers directly above it. The row for n=5 gives the coefficients of (1+x)^5 immediately.
每一行以 1 开头并以 1 结尾，中间的每个数都等于其正上方两个数之和。n=5 的一行直接给出了 (1+x)^5 的系数。
For larger values of n, the nCr formula is more efficient than writing out Pascal’s triangle. However, if n is small, the triangle can save time under exam pressure.
对于较大的 n，使用 nCr 公式比逐行写出杨辉三角更高效。不过，当 n 较小时，三角形可以帮助你在考试压力下节省时间。
6. Worked Example: Expanding (1+x)⁵ | 例题：展开 (1+x)⁵
Use the coefficients from Pascal’s triangle for n=5: 1, 5, 10, 10, 5, 1. These represent C(5,0), C(5,1), C(5,2), C(5,3), C(5,4) and C(5,5).
使用 n=5 时杨辉三角的系数：1、5、10、10、5、1。这些系数分别代表 C(5,0)、C(5,1)、C(5,2)、C(5,3)、C(5,4) 和 C(5,5)。
(1+x)⁵ = 1 + 5x + 10x² + 10x³ + 5x⁴ + x⁵
Notice that the powers of x increase from 0 to 5, while the coefficients are symmetric. This symmetry appears because C(5,r) = C(5,5−r).
注意 x 的幂从 0 增加到 5，而系数呈对称分布。这种对称性来源于 C(5,r) = C(5,5−r)。
If you substitute x = 1, the left-hand side becomes 2⁵ = 32, and the right-hand side becomes 1 + 5 + 10 + 10 + 5 + 1 = 32. This verifies the expansion.
如果代入 x = 1，左边等于 2⁵ = 32，右边等于 1 + 5 + 10 + 10 + 5 + 1 = 32，这验证了展开式的正确性。
7. Worked Example: (1 + 3x)⁸ Up to x³ | 例题：展开 (1 + 3x)⁸ 至 x³
For (1 + 3x)⁸, replace x in the standard formula by 3x. We need only the terms up to x³:
对于 (1 + 3x)⁸，需要把标准公式中的 x 替换为 3x。我们只需要直到 x³ 的项：
(1 + 3x)⁸ = 1 + 8(3x) + C(8,2)(3x)² + C(8,3)(3x)³ + …
Calculate the coefficients carefully:
仔细计算各项系数：
1 + 24x + 28(9x²) + 56(27x³) + …
= 1 + 24x + 252x² + 1512x³ + …
A very common mistake is to write C(8,2)x² but forget to square the 3 in 3x. The factor 3 must be raised to the same power as x in each term.
一个非常常见的错误是写出 C(8,2)x² 却忘记对 3x 中的 3 取平方。3 的幂必须与每一项中 x 的幂保持一致。
8. The Binomial Series for Rational n | 有理数 n 的二项式级数
A-Level Mathematics also requires the expansion of (1+x)^n when n is negative or a fraction. In these cases the expansion is an infinite series:
A-Level 数学还要求掌握 n 为负数或分数时 (1+x)^n 的展开。此时展开式为一个无穷级数：
(1+x)^n = 1 + nx + n(n−1)/2! x² + n(n−1)(n−2)/3! x³ + …
This series is often called the Binomial Series. It is valid only when |x| < 1, unless n is a non-negative integer.
这个级数通常被称为二项式级数。除非 n 是非负整数，否则它只在 |x| < 1 时有效。
For example, when n = −1, the formula gives:
例如，当 n = −1 时，公式给出：
1/(1+x) = 1 − x + x² − x³ + x⁴ − …
When n = 1/2, the first few terms of the expansion for √(1+x) are:
当 n = 1/2 时，√(1+x) 的展开前几项为：
√(1+x) = 1 + (1/2)x − (1/8)x² + (1/16)x³ − …
9. The Validity Condition |x| < 1 | 收敛条件 |x| < 1
When n is not a non-negative integer, the binomial series is infinite. An infinite series does not always have a finite sum, and the binomial series must satisfy |x| < 1 to converge.
当 n 不是非负整数时，二项式级数是无穷级数。无穷级数并不总是具有有限和，二项式级数必须满足 |x| < 1 才能收敛。
If |x| ≥ 1, the terms typically grow without bound, so the series is not valid in ordinary real arithmetic.
如果 |x| ≥ 1，各项通常会无限增大，因此该级数在普通实数运算中不成立。
In exam questions, you will often be asked to state the range of values for which the expansion is valid. For (1 + a x)^n, the condition is |a x| < 1, which is equivalent to |x| < 1/|a|.
在考试中，常会要求你写出展开式成立的取值范围。对于 (1 + a x)^n，条件是 |a x| < 1，等价于 |x| < 1/|a|。
For positive integer n, this restriction is not needed because the expansion is finite and always converges.
当 n 为正整数时，不需要这一限制，因为展开式是有限项，必然收敛。
10. Adapting the Formula to (1 + a x)ⁿ | 将公式推广到 (1 + a x)ⁿ
Many exam questions involve an expression such as (1 + 2x)⁻¹ or (1 − x/3)^(1/2). The most reliable method is to replace x in the standard series by the whole expression inside the bracket.
许多考试题目涉及形如 (1 + 2x)⁻¹ 或 (1 − x/3)^(1/2) 的表达式。最可靠的方法是把括号内的整个表达式代入标准公式中的 x。
(1 + a x)^n = 1 + n(a x) + n(n−1)/2! (a x)² + n(n−1)(n−2)/3! (a x)³ + …
The validity condition becomes |a x| < 1. The table below illustrates common substitutions:
收敛条件变为 |a x| < 1。下表展示了常见的代换方式：
| Expression 表达式 | Value of a a 的值 | Validity 收敛范围 |
|---|---|---|
| (1 + 2x)⁻¹ | a = 2 | |x| < 1/2 |
| (1 − 3x)^(1/2) | a = −3 | |x| < 1/3 |
| (1 − x/4)⁻² | a = −1/4 | |x| < 4 |
Remember to simplify each term fully. For example, (1 − 3x)^(1/2) contains alternating signs because a is negative.
请记住要把每一项化简完整。例如，(1 − 3x)^(1/2) 因为 a 为负值，所以展开式会呈现正负号交替。
11. Common Mistakes and Exam Tips | 常见错误与考试技巧
Students frequently lose marks by forgetting to square or cube the constant factor inside the bracket. When expanding (1 + 3x)⁸, the term in x² must include 3² = 9.
学生经常因为忘记对括号内的常数因子取平方或立方而失分。在展开 (1 + 3x)⁸ 时，x² 项必须包含 3² = 9。
Another common error is using the wrong sign. When the expression is (1 − 2x)^n, the terms alternate because each power of −2x contributes a negative sign when raised to an odd power.
另一个常见错误是弄错正负号。当表达式为 (1 − 2x)^n 时，由于 −2x 的奇数次幂会贡献负号，因此展开式中各项正负交替出现。
For negative or fractional n, you must state the range of validity. If the question says “state the range of values for which this expansion is valid,” write |x| < 1/|a| explicitly.
对于负指数或分数指数，你必须写出收敛范围。如果题目要求“写出该展开式成立时 x 的取值范围”，要明确写出 |x| < 1/|a|。
Use your calculator to check one numerical value. For example, in the expansion of (1+x)⁵ with x = 0.2, both sides should give approximately 2.48832.
可以用计算器代入一个数值检验。例如，在 (1+x)⁵ 中取 x = 0.2，两边都应给出约等于 2.48832。
Finally, read the question carefully. If it asks for “the first four terms,” do not write the entire expansion. If it asks for “the coefficient of x³,” give only that coefficient and not the full term.
最后，务必仔细审题。如果题目要求“前四项”，就不要写出完整展开。如果题目要求“x³ 的系数”，就只给出该系数，而不用写出完整项。
12. Practice Questions and Final Advice | 练习题目与总结建议
To master the binomial expansion of (1+x)^n, practice is essential. Try these two questions:
掌握 (1+x)^n 的二项式展开离不开练习。试着解决以下两个问题：
Question 1: Expand (1 − 2x)⁻² up to the term in x³, and state the range of validity.
问题 1：将 (1 − 2x)⁻² 展开至含 x³ 的项，并写出其收敛范围。
Answer guidance: Using the binomial series with n = −2 and a = −2 gives 1 + 4x + 12x² + 32x³ + … with validity |x| < 1/2.
答案提示：使用 n = −2、a = −2 的二项式级数，可得 1 + 4x + 12x² + 32x³ + …，收敛范围为 |x| < 1/2。
Question 2: Find the coefficient of x⁴ in the expansion of (1 + x/2)¹².
问题 2：求 (1 + x/2)¹² 的展开式中 x⁴ 的系数。
Answer guidance: The required term is C(12,4)(x/2)⁴ = 495 × x⁴/16, so the coefficient is 495/16.
答案提示：所需项为 C(12,4)(x/2)⁴ = 495 × x⁴/16，因此系数为 495/16。
No matter how the question is phrased, always identify n first, then decide whether the expansion is finite or infinite, and finally apply the correct formula. A systematic approach will reduce errors and earn full marks.
无论题目如何表述，都要先确定 n，再判断展开式是有限项还是无穷级数，最后套用正确的公式。按部就班的解题方法能够减少错误并帮助获得满分。
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