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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 \[\left( \frac{8}{2} + 1 \right)^{th}\] i.e., 5th term is the middle term.
It is given that the middle term is 1120.
\[\therefore T_5 = 1120\]
\[ \Rightarrow^8 C_4 \left( \frac{p}{2} \right)^{8 - 4} \left( 2 \right)^4 = 1120\]
\[ \Rightarrow p^4 = 16\]
\[ \Rightarrow p = \pm 2\]
Hence, the real values of p is \[\pm 2\]
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