Problem 55 - Algebra and Trigonometry

Question

Find the middle term in the expansion of \(\left(\frac{3}{x}+\frac{x}{3}\right)^{10}\)

Step-by-Step Solution

Answer

The middle terms of the binomial expansion \(\left(\frac{3}{x} + \frac{x}{3}\right)^{10}\) are \(T_6\) and \(T_7\) which can be calculated using the appropriate formula.

Step 1: Identify parts of the binomial

In the binomial \( \left(\frac{3}{x} + \frac{x}{3} \right)^{10} \), 'a' is \(\frac{3}{x}\) and 'b' is \(\frac{x}{3}\). The number of terms is 10 + 1 = 11.

Step 2: Calculate the terms

Now, we must find the 6th and 7th term of the binomial expansion since they are the middle terms. Using the formula \(T_{r+1} = ^nC_r(a^{n-r})(b^r)\), we calculate 6th term as \(T_6 = ^{10}C_5(\frac{3}{3} (\frac{3}{x})^{(10-5)}) (\frac{x}{3})^{5}\) and similarly 7th term as \(T_7 = ^{10}C_6(\frac{3}{3} (\frac{3}{x})^{(10-6)}) (\frac{x}{3})^{6}\)

Step 3: Simplify the results

Simplifying both the terms we get the answer.