Problem 55 - College Algebra

Question

Find the middle term in the expansion of $$\left(\frac{3}{x}+\frac{x}{3}\right)^{16}$$.

Step-by-Step Solution

Answer

The middle term in the expansion of the given binomial expression is 12870.

Step 1: Identify the coefficients

First, we are dealing with the binomial theorem. Let's identify the coefficients \(a = \frac{3}{x}\) and \(b = \frac{x}{3}\)

Step 2: Find the index of the middle term

For the given expression, the total number of terms would be \(n+1 = 16+1 = 17\). This is odd so the middle term is the \(\frac{17+1}{2} = 9^{th}\) term in the expansion.

Step 3: Compute the middle term

To find the middle term we use the formula: \( _{n}C_{r} . a^{n-r} . b^{r} \). So, substituting \(n = 16\), \(r= 8\), \(a = \frac{3}{x}\) and \(b = \frac{x}{3}\) gives us \(_{16}C_{8} . \left(\frac{3}{x}\right)^{16-8} . \left(\frac{x}{3}\right)^{8} = _{16}C_{8} . x^{8} . \left(\frac{3}{x}\right)^{8}\). Simplifying gives us \( _{16}C_{8}\).

Step 4: Evaluate _{n}C_{r}

Finally, the ninth term of the binomial expression is found by evaluating _{16}C_{8} = 12870. Therefore, the middle term in the expansion is 12870.

Key Concepts

Coefficient Expansion

The Binomial Theorem allows us to expand expressions of the form \((a+b)^n\). In this theorem, each term in the expansion involves a specific coefficient known as the binomial coefficient. This coefficient dictates the importance or weight of each individual term within the expansion.

The formula for any term in the expansion is given by:\[_{n}C_{r} \cdot a^{n-r} \cdot b^{r}\]Where \(_{n}C_{r}\) is the binomial coefficient, \(a\) and \(b\) are the terms from the original binomial expression, \(n\) is the exponent to which the binomial is raised, and \(r\) is the term number we want to find (starting from 0).

• At \(r=0\), the coefficient is \(_{n}C_{0}\).
• For subsequent terms, the coefficient is \(_{n}C_{1}\), \(_{n}C_{2}\), ..., \(_{n}C_{n}\).

As seen from the exercise, for the middle term in an expansion, the binomial coefficient \(_{16}C_{8}\) becomes essential in determining our result of 12870.

Middle Term Formula

The middle term in a binomial expansion is significant, especially when the number of terms is odd. It serves as the central point of symmetry for the expanded polynomial.

For a binomial expression \((a+b)^n\), the number of terms produced is \(n+1\). So, when \(n+1\) is odd, the middle term is considered to be the \((\frac{n+1}{2})^{th}\) term. In our specific case, with \(n=16\), there are 17 terms in total.

• This value of 17 is odd, making the middle term fall in the \((\frac{17}{2}) = 9^{th}\) position.
• Finding this middle term is crucial in problems that deal solely in evaluating or simplifying binomial expansions.

The middle term, as calculated, often showcases the most central dynamic of the expansion, like in this exercise where its answer was 12870 and provided the pivotal point of the expression.

Combination Formula

Often in mathematics, we encounter terms like \(_{n}C_{r}\), which involve combinations. This formula is fundamental when managing binomial expansions, as it tells how many ways you can select \(r\) elements from \(n\) elements, without considering order.

The combination formula is expressed as:\[_{n}C_{r} = \frac{n!}{r!(n-r)!}\]where \(n!\) represents the factorial of \(n\). This factorial operation involves multiplying the given number \(n\) by each preceding number down to one.

• In our exercise, we calculated \(_{16}C_{8}\) which involves determining how many ways we can select 8 elements from a pool of 16.
• This calculation provides the crucial coefficient for the middle term, further verifying its value as 12870.

Understanding how to compute combinations is pivotal in successfully applying the binomial theorem, allowing us to determine any term's coefficient efficiently.