Find the term indicated in each expansion. (x − 1)9; fifth term

Ch. 8 - Sequences, Induction, and Probability

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Blitzer 8th Edition

Ch. 8 - Sequences, Induction, and Probability

문제 41

9장, 문제 41

Find the term indicated in each expansion. (x − 1)⁹; fifth term

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Identify the general term formula for the binomial expansion of \((x - 1)^9\). The \(k\)-th term is given by \(T_{k+1} = \binom{9}{k} x^{9-k} (-1)^k\) where \(k\) starts from 0.

Since we want the fifth term, set \(k = 4\) because the first term corresponds to \(k=0\).

Substitute \(k=4\) into the general term formula: \(T_5 = \binom{9}{4} x^{9-4} (-1)^4\).

Calculate the binomial coefficient \(\binom{9}{4}\) using the formula \(\binom{n}{r} = \frac{n!}{r!(n-r)!}\), but do not compute the numerical value yet.

Write the fifth term as \(T_5 = \binom{9}{4} x^5 (1)\) since \((-1)^4 = 1\), and this expression represents the fifth term in the expansion.

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주요 개념

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Binomial Theorem

The Binomial Theorem provides a formula to expand expressions of the form (a + b)^n. It states that the expansion is the sum of terms involving binomial coefficients, powers of a, and powers of b. This theorem is essential for finding specific terms in binomial expansions.

Binomial Coefficients

Binomial coefficients, denoted as C(n, k) or "n choose k," represent the number of ways to choose k elements from n. They appear as coefficients in the binomial expansion and can be calculated using factorials or Pascal's Triangle. These coefficients determine the weight of each term.

Term Identification in Binomial Expansion

In the expansion of (x + y)^n, the (k+1)-th term is given by C(n, k) * x^(n-k) * y^k. To find the fifth term, substitute k = 4. Understanding this formula allows you to directly compute any specific term without expanding the entire expression.