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.

Answer 1

In the binomial expansion of ${\left(\frac{p}{2}+2\right)}^8$, we observe that ${\left(\frac{8}{2}+1\right)}^{\mathrm{th}}$ i.e., 5^th term is the middle term.

It is given that the middle term is 1120.

$$\begin{gathered} \therefore T_5=1120 \\ {\Rightarrow }^8{C}_4{\left(\frac{p}{2}\right)}^{8-4}{\left(2\right)}^4=1120 \\ \Rightarrow p^4=16 \\ \Rightarrow p=\pm 2 \end{gathered}$$

Hence, the real values of p is $\pm 2$.