In Exercises 39–48, find the term indicated in each expansion. (x − 1/2)^9; fourth term

Verified step by step guidance

1. insert step 1> Identify the general term in the binomial expansion of (x - \frac{1}{2})^9 using the Binomial Theorem, which is given by T_{k+1} = \binom{n}{k} a^{n-k} b^k.
2. insert step 2> In this problem, n = 9, a = x, and b = -\frac{1}{2}. We are looking for the fourth term, so k = 3.
3. insert step 3> Substitute the values into the general term formula: T_4 = \binom{9}{3} x^{9-3} (-\frac{1}{2})^3.
4. insert step 4> Calculate the binomial coefficient \binom{9}{3}, which represents the number of ways to choose 3 elements from a set of 9.
5. insert step 5> Simplify the expression by calculating x^{6} and (-\frac{1}{2})^3, and then multiply all parts together to find the fourth term.

Key Concepts

Binomial Theorem

The Binomial Theorem provides a formula for expanding expressions of the form (a + b)^n. It states that the expansion can be expressed as a sum of terms involving binomial coefficients, which are calculated using combinations. Each term in the expansion is of the form C(n, k) * a^(n-k) * b^k, where C(n, k) is the binomial coefficient, a and b are the terms being raised to the power n, and k is the term index.

Binomial Coefficients

Binomial coefficients, denoted as C(n, k) or 'n choose k', represent the number of ways to choose k elements from a set of n elements without regard to the order of selection. They are calculated using the formula C(n, k) = n! / (k!(n-k)!), where '!' denotes factorial. In the context of the Binomial Theorem, these coefficients determine the weight of each term in the expansion.

Term Index in Binomial Expansion

In a binomial expansion, each term can be indexed starting from zero. The k-th term in the expansion of (a + b)^n is given by the formula C(n, k) * a^(n-k) * b^k. To find a specific term, such as the fourth term, one must identify the correct index (k = 3 for the fourth term) and substitute it into the formula along with the values of a, b, and n.