Question

The middle term in the expansion of ${\left[x-\frac{1}{x}\right]}^{18}$ is

• A. ${}^{18}C_9$
• B. $-{}^{18}C_9$
• C. ${}^{18}C_{10}$
• D. $-{}^{18}C_{10}$

Answer 1

The correct option is B

$-{}^{18}C_9$

Explanation for the correct option:

Compute the middle term of the given expansion.

An expansion ${\left[x-\frac{1}{x}\right]}^{18}$ is given.

The number of terms in the expansion is $18+1=19$.

The middle term of the expansion is given by $\frac{19+1}{2}=10$. That is $T_{10}$ is the middle term of the given expansion.

Compute the value of $T_{10}$ as follows:

$$\begin{gathered} T_{10}={}^{18}C_9·{\left(x\right)}^9{\left(-\frac{1}{x}\right)}^{18-9} \\ \Rightarrow T_{10}=-{}^{18}C_9·{\left(x\right)}^9\left(\frac{1}{x^9}\right) \\ \Rightarrow T_{10}=-{}^{18}C_9 \end{gathered}$$

Therefore, the middle term of the given expansion is $-{}^{18}C_9$.

Hence, option $B$ is the correct answer.