First slide

Binomial theorem for positive integral Index

Question

If $a$ is a real number and if the middle term of ${(\frac{a}{3} + 3)}^{8}$ is 1120, then value of $a$ is

Moderate

A

$\pm 2$
B

$\pm 1$
C

$\pm \sqrt{3}$
D

$\pm \sqrt{2}$

Solution

Middle term of ${(\frac{a}{3} + 3)}^{8}$ is the $(\frac{8}{2} + 1)$ th or $5^{t h}$ term. We have
$T_{5} = T_{4 + 1} =^{8} C_{4} {(\frac{a}{3})}^{4} (3)^{4} = 70 a^{4}$
We are given $T_{5} = 1120 \Rightarrow 70 a^{4} = 1120$
$\Rightarrow a^{4} = 16 = 2^{4} \Rightarrow a = \pm 2$

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