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

Ch. 8 - Sequences, Induction, and Probability

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Blitzer 8th Edition

Ch. 8 - Sequences, Induction, and Probability

문제 55

9장, 문제 55

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

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Identify the binomial expression and the exponent: the expression is \(\left( \frac{3}{x} + \frac{x}{3} \right)^{10}\), where \(n = 10\).

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 \(n\).

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 \(k = 5\)).

Substitute \(a = \frac{3}{x}\), \(b = \frac{x}{3}\), \(n = 10\), and \(k = 5\) into the general term formula: \(T_6 = \binom{10}{5} \left( \frac{3}{x} \right)^{10-5} \left( \frac{x}{3} \right)^5\).

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.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.