Question

Find the middle terms(s) in the expansion of:

(v) \[\left( x - \frac{1}{x} \right)^{2n + 1}\]

 

Answer 1

\[\left( x - \frac{1}{x} \right)^{2n + 1} \]
\[\text{ Here, } \left( 2n + 1 \right) \text{ is an odd number .}  \]
\[\text{ Therefore, the middle terms are } \left( \frac{2n + 1 + 1}{2} \right)\text{ th and} \left( \frac{2n + 1 + 1}{2} + 1 \right)\text{ th i . e . (n + 1)th and (n + 2)th terms }  . \]
\[\text{ Now, we have: }  \]
\[ T_{n + 1} \]
\[ =^{2n + 1}{}{C}_n x^{2n + 1 - n} \times \frac{( - 1 )^n}{x^n}\]
`=(-1)^n  "^(2n+1)C_n x`
\[\text{ And,}  \]
\[ T_{n + 2} = T_{n + 1 + 1} \]
\[ =^{2n + 1}{}{C}_{n + 1} x^{2n + 1 - n - 1} \frac{( - 1 )^{n + 1}}{x^{n + 1}}\]
\[ = ( - 1 )^{n + 1} \]  ` "^{2n + 1}C_{n + 1} \times \frac{1}{x}`