First slide

Binomial theorem for positive integral Index

Question

If p is a real number and if the middle tern in the expansion of ${(\frac{p}{2} + 2)}^{8}$ is 1120,then the value of p is

Moderate

A

$\pm 3$
B

$\pm 1$
C

$\pm 2$
D

None of these

Solution

Given expression is ${(\frac{p}{2} + 2)}^{8}$
Here, n =8 (even)
$\begin{array}{l} \Rightarrow Middle term = (\frac{8}{2} + 1) t h = 5 th \\ \begin{aligned} T_{5} & =^{8} C_{4} {(\frac{p}{2})}^{8 - 4} (2)^{4} \\ \Rightarrow & \frac{8 \times 7 \times 6 \times 5}{4 \times 3 \times 2 \times 1} \times \frac{p^{4}}{2^{4}} \times 2^{4} & = 1120 \Rightarrow P^{4} = 16, p = \pm 2 \end{aligned} \end{array}$

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