Question:
If $p$ is a real number and if the middle term in the expansion of $\left(\frac{p}{2}+2\right)^{8}$ is 1120, find $p$.
Solution:
In the binomial expansion of $\left(\frac{p}{2}+2\right)^{8}$, we observe that $\left(\frac{8}{2}+1\right)^{\text {th }}$ i.e., $5^{\text {th }}$ term is the middle term.
It is given that the middle term is 1120.
$\therefore T_{5}=1120$
$\Rightarrow^{8} C_{4}\left(\frac{p}{2}\right)^{8-4}(2)^{4}=1120$
$\Rightarrow p^{4}=16$
$\Rightarrow p=\pm 2$
Hence, the real values of $p$ is $\pm 2$.