Find the middle term in the expansion of (3/x + x/3)^10

Verified step by step guidance

1. Identify the general term in the binomial expansion of \((a + b)^n\), which is given by \(T_{k+1} = \binom{n}{k} a^{n-k} b^k\).
2. For the expression \((3/x + x/3)^{10}\), set \(a = 3/x\), \(b = x/3\), and \(n = 10\).
3. The middle term in a binomial expansion occurs at \(k = \frac{n}{2}\) when \(n\) is even. Here, \(n = 10\), so \(k = 5\).
4. Substitute \(k = 5\) into the general term formula: \(T_{6} = \binom{10}{5} \left(\frac{3}{x}\right)^{10-5} \left(\frac{x}{3}\right)^5\).
5. Simplify the expression for \(T_6\) by calculating \(\binom{10}{5}\) and simplifying the powers of \(x\) and constants.

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 varies from 0 to n.

Middle Term in Binomial Expansion

In a binomial expansion of (a + b)^n, the middle term can be found based on whether n is even or odd. If n is even, the middle term is the (n/2 + 1)th term; if n is odd, there are two middle terms at positions (n/2) and (n/2 + 1). For n = 10, which is even, the middle term will be the 6th term in the expansion, calculated using the Binomial Theorem.

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 binomial expansion, these coefficients determine the weight of each term in the expansion and are crucial for finding specific terms, such as the middle term.