Find the term indicated in each expansion. (x − 1/2)9; fourth term

Ch. 8 - Sequences, Induction, and Probability

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Blitzer 8th Edition

Ch. 8 - Sequences, Induction, and Probability

Problem 45

Chapter 9, Problem 45

Find the term indicated in each expansion. (x − 1/2)⁹; fourth term

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Recall the Binomial Theorem, which states that the expansion of $(a + b)^n$ is given by the sum of terms: $\binom{n}{k} a^{n-k} b^k$, where $k$ ranges from 0 to $n$.

Identify the values in the problem: $a = x$, $b = -\frac{1}{2}$, and $n = 9$.

Note that the term number $r$ in the expansion corresponds to $k = r - 1$. Since we want the fourth term, set $k = 3$.

Write the general form for the fourth term: $T_4 = \binom{9}{3} x^{9-3} \left(-\frac{1}{2}\right)^3$.

Simplify the powers and coefficients as much as possible (without calculating the final numeric value) to express the fourth term clearly.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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 multiplied by powers of a and 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

Each term in a binomial expansion corresponds to a specific value of k, starting from k=0 for the first term. The (r)th term is given by C(n, r-1) * a^(n-(r-1)) * b^(r-1). Understanding this indexing helps locate and calculate the exact term requested, such as the fourth term in the expansion.

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