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

Verified step by step guidance

1. Identify the binomial expression and the exponent: the expression is \(\left( \frac{3}{x} + \frac{x}{3} \right)^{10}\), where \(n = 10\).
2. Recall the general term in the binomial expansion of \((a + b)^n\) is given by \(T_{k+1} = \binom{n}{k} a^{n-k} b^k\), where \(k\) ranges from 0 to \(n\).
3. Determine the middle term: since \(n = 10\) is even, the middle term is the \(\left( \frac{n}{2} + 1 \right)\)-th term, which is the 6th term (where \(k = 5\)).
4. Substitute \(a = \frac{3}{x}\), \(b = \frac{x}{3}\), \(n = 10\), and \(k = 5\) 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 powers and combine like terms carefully, paying attention to the exponents of \(x\) and the constants, to express the middle term in its simplest form.

Key Concepts

Binomial Theorem

The Binomial Theorem provides a formula to expand expressions of the form (a + b)^n into a sum involving terms with binomial coefficients. Each term is given by C(n, k) * a^(n-k) * b^k, where C(n, k) is the combination of n items taken k at a time.

Identifying the Middle Term in a Binomial Expansion

For an expansion of (a + b)^n, the total number of terms is n + 1. If n is even, the middle term is the (n/2 + 1)-th term. This term can be found by substituting k = n/2 into the general term formula.

Handling Algebraic Expressions with Variables and Exponents

When expanding expressions like (3/x + x/3)^10, it is important to carefully apply exponent rules to variables and constants. Simplifying powers of variables with negative or fractional exponents ensures the correct form of each term.