If p is a real number and if the middle term in the expansion of

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 $\pm 2$.

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