Question

Find the middle term in the expansion of ${\displaystyle {(\frac{x}{\text{a}}-\frac{\text{a}}{x})}^{10}}$

Answer 1

Here, a = ${\displaystyle \frac{x}{\text{a}}}$, b = ${\displaystyle -\frac{\text{a}}{x}}$, n = 10

Now, n is even

∴ ${\displaystyle \frac{\text{n}+2}{2}=\frac{10+2}{2}}$ = 6

∴ Middle term is t₆, for which r = 5.

We have, t_r+1 = ⁿC_r a^n−r .b^r

t₆ = ${\displaystyle {}^{10}{\text{C}}_5.{\left(\frac{x}{\text{a}}\right)}^5.{(-\frac{\text{a}}{x})}^5}$

= ¹⁰C₅ × −1

= ${\displaystyle \frac{-10!}{5!5!}=-\frac{10\times 9\times 8\times 7\times 6}{5\times 4\times 3\times 2\times 1}}$

= − 252

∴ Middle term is − 252.