Power of at least one term decreased by 1. (Binomial Expansion) | 二项式展开中至少一项的幂次减1

📚 Power of at least one term decreased by 1. (Binomial Expansion) | 二项式展开中至少一项的幂次减1

When expanding a binomial expression (a + b)ⁿ, a fascinating pattern governs the powers of the variables. As you move from one term to the next, the exponent of the first variable a is reduced by exactly 1, while the exponent of the second variable b increases by 1. This means that in every step, the power of at least one term is decreased by 1. Understanding this simple observation unlocks a deeper appreciation of binomial expansions, term-finding strategies, and even the behaviour of infinite series.

在展开二项式 (a + b)ⁿ 时，各项的变量指数遵循一个奇妙的规律。从一项移动到下一项时，第一个变量 a 的指数每次都精确地减少 1，同时第二个变量 b 的指数增加 1。这意味着每一步中，至少有一项的幂次减少了 1。理解这一简单观察有助于深刻领会二项式展开、项的查找技巧乃至无穷级数的行为。

1. What is the Binomial Theorem? | 二项式定理是什么？

The Binomial Theorem provides a formula for expanding expressions of the form (a + b)ⁿ where n is a positive integer. The expansion is a sum of terms involving powers of a and b, each multiplied by a binomial coefficient. For example, (a + b)² = a² + 2ab + b². The theorem generalises this pattern for any natural number n.

二项式定理给出了展开形如 (a + b)ⁿ（n 为正整数）的表达式的公式。展开式是一系列包含 a 和 b 的幂的项之和，每项乘以一个二项式系数。例如 (a + b)² = a² + 2ab + b²。该定理将这一模式推广到任意自然数 n。

---

2. The General Term and Index Changes | 通项与指数变化

The general term in the expansion of (a + b)ⁿ is given by T_r+1 = ⁿC_r a^n-r b^r, where r runs from 0 to n. Notice that the power of a is n − r, and the power of b is r. When we increase r by 1 to get the next term, the exponent of a decreases by 1 while the exponent of b increases by 1. Thus, at every step, the power of at least one term – in fact the power of a – is decreased by 1.

(a + b)ⁿ 展开式的通项为 T_r+1 = ⁿC_r a^n-r b^r，其中 r 从 0 取到 n。注意 a 的指数是 n − r，b 的指数是 r。当我们把 r 增加 1 得到下一项时，a 的指数减 1，而 b 的指数加 1。因此，每一步中至少有一项的幂次——实际上就是 a 的幂次——减少了 1。

---

3. Why "Power of at Least One Term Decreased by 1"? | 为什么说“至少一项的幂次减1”？

The phrase highlights the shift in exponents between successive terms. Consider two adjacent terms: T_r+1 = ⁿC_r a^n-r b^r and T_r+2 = ⁿC_r+1 a^n-r-1 b^r+1. The exponent of a has fallen by 1, whereas the exponent of b has risen by 1. Even if we view the entire term as a single entity, the variable part undergoes a net change where one variable loses a unit of power. This is the meaning behind "power of at least one term decreased by 1".

这句话突出了相邻项之间指数的转移。考虑两个相邻的项：T_r+1 = ⁿC_r a^n-r b^r 和 T_r+2 = ⁿC_r+1 a^n-r-1 b^r+1。a 的指数降了 1，而 b 的指数升了 1。即使将整个项视为一个整体，其变量部分也发生了净变化，其中一个变量丧失了一单位幂次。这就是“至少一项的幂次减1”背后的含义。

---

4. Example: Expanding (x + y)⁴ | 示例：展开 (x + y)⁴

Let's expand (x + y)⁴ using the binomial theorem: (x + y)⁴ = ⁴C₀ x⁴y⁰ + ⁴C₁ x³y¹ + ⁴C₂ x²y² + ⁴C₃ x¹y³ + ⁴C₄ x⁰y⁴ = x⁴ + 4x³y + 6x²y² + 4xy³ + y⁴ ecreasing by 1 each time. The power of y increases by 1 each time. Therefore, in every transition the power of at least one term (the x-term) is decreased by 1.

让我们用二项式定理展开 (x + y)⁴：(x + y)⁴ = ⁴C₀ x⁴y⁰ + ⁴C₁ x³y¹ + ⁴C₂ x²y² + ⁴C₃ x¹y³ + ⁴C₄ x⁰y⁴ = x⁴ + 4x³y + 6x²y² + 4xy³ + y⁴。这里 x 的指数依次为 4 → 3 → 2 → 1 → 0，每次都减 1。y 的指数每次都加 1。因此，每次过渡中至少有一项（x 对应的项）的幂次减少了 1。

---

广告 / Sponsored

5. The Form (1 + x)ⁿ and Power Reduction | (1+x)ⁿ 形式与幂次减少

Often we deal with expansions of (1 + x)ⁿ. The general term is ⁿC_r 1^n-r x^r = ⁿC_r x^r. Although the factor 1^n-r is always 1, its exponent still implicitly decreases by 1 each time r increases. Hence the statement "power of at least one term decreased by 1" remains true: the exponent of the constant term (treating 1 as a variable) drops by 1 while the exponent of x rises by 1. This way of thinking helps when later dealing with (a + bx)ⁿ where the same pattern applies to the first part a.

我们经常处理 (1 + x)ⁿ 的展开式。其通项为 ⁿC_r 1^n-r x^r = ⁿC_r x^r。虽然因子 1^n-r 恒等于 1，但其指数在 r 增加时仍然隐性地每次减少 1。因此“至少一项的幂次减1”的陈述依然成立：常数项（把 1 视为变量）的指数下降 1，而 x 的指数上升 1。这种思考方式有助于后续处理 (a + bx)ⁿ，其中第一个部分 a 也遵循同样的模式。

---

6. Using the Pattern to Find a Specific Term | 利用规律寻找特定项

Knowing that one variable's power decreases by 1 with each term provides a quick way to locate a desired term. For instance, in the expansion of (2x + 3)⁸, suppose we want the term containing x⁵ is 8 − r. Setting 8 − r = 5 gives r = 3. Thus the required term is the 4th term (r = 3), T₄ = ⁸C₃ (2x)⁵ (3)³.

知道一个变量的幂次每项减少 1 提供了快速定位所求项的方法。例如，在 (2x + 3)⁸ 的展开式中，假设我们想找包含 x⁵ 的项。由于第一个变量的指数从 8 开始，每一步减 1，经过 r 步后指数为 8 − r。令 8 − r = 5 得 r = 3。因此所求项是第 4 项（r = 3），T₄ = ⁸C₃ (2x)⁵ (3)³。

---

7. Extension to Negative and Fractional Powers | 扩展到负指数与分数指数

When n is negative or a fraction, the binomial expansion becomes an infinite series, but the index pattern persists. For (1 + x)ⁿ, the expansion is 1 + nx + [n(n−1)/2!] x² + ... . The exponent of x increases by 1 in successive terms, meaning that the power of the implicit constant term (1^n−r) decreases by 1 each time. Even though the series never ends, the 'power of at least one term decreased by 1' rule continues to hold, linking finite and infinite expansions.

当 n 为负数或分数时，二项式展开变成一个无穷级数，但指数模式依然保持。对于 (1 + x)ⁿ，展开式为 1 + nx + [n(n−1)/2!] x² + ... 。相邻项中 x 的指数每次增加 1，这意味着隐含的常数项（1^n−r）的指数每次减少 1。尽管级数永无止境，但“至少一项的幂次减1”的规则依然成立，连接了有限与无限展开。

---

8. Summary | 总结

In every binomial expansion, moving from one term to the next always causes the power of at least one variable to decrease by 1—the first variable in (a + b)ⁿ loses one unit of its exponent while the second gains one. This simple observation is the key to writing general terms, finding specific coefficients, and understanding the structure of both finite and infinite expansions. Remember: whichever form the binomial takes, the dance of the exponents never changes: at least one power drops by 1, every single time.

在每一个二项式展开中，从一项移到下一项总会导致至少一个变量的幂次减少 1——在 (a + b)ⁿ 中，第一个变量丢失一单位指数而第二个变量获得一单位。这一简单观察是写出通项、寻找特定系数以及理解有限和无限展开结构的关键。请记住：无论二项式以何种形式出现，指数的舞步永不改变——每一次，至少有一个幂次下降 1。

---

Published by TutorHao | Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）