Question

Find the middle term in the expansion of ${(\frac{x}{a}-\frac{a}{x})}^{21}$

• A. ${{}^{20}C}_{10}\frac{x}{a},{{}^{21}C}_{10}\frac{a}{x}$
• B. ${{}^{20}C}_9\frac{x}{a},{{}^{21}C}_{10}\frac{a}{x}$
• C. ${{}^{21}C}_{10}\frac{x}{a},-{{}^{21}C}_{10}\frac{a}{x}$
• D. ${{}^{21}C}_9\frac{x}{a},{{}^{21}C}_{10}\frac{a}{x}$

Answer 1

The correct option is C ${{}^{21}C}_{10}\frac{x}{a},-{{}^{21}C}_{10}\frac{a}{x}$

There are 22 terms in the expansion of $(\frac{x}{a}-\frac{a}{x}{)}^{21}$.
So, the middle terms are $T_{11}$ and $T_{12}$
$T_{11}{=}^{21}{\text{C}}_{10}(\frac{x}{a}{)}^{11}(\frac{-a}{x}{)}^{10}{=}^{21}{\text{C}}_{10}(\frac{x}{a})$
and
$T_{12}{=}^{21}{\text{C}}_{11}(\frac{x}{a}{)}^{10}(\frac{-a}{x}{)}^{11}={-}^{21}{\text{C}}_{11}(\frac{a}{x})={-}^{21}{\text{C}}_{10}(\frac{a}{x})$