If a is positive real number and if the middle term of $$a3+38is$$ $$1120$$ then value of a is

Middle term of $$a3+38is$$ the $$82+1th$$ $$or5th$$ term. We have $$T5=T4+1=8C4a34(3)4=70a4We$$ are given $$T5=1120$$⇒$$70a4=1120$$⇒ $$a4=16=24$$⇒$$a=$$±$$2$$

Binomial theorem for positive integral Index

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Question

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

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Solution

Middle term of ${(\frac{a}{3} + 3)}^{8}$ is the $(\frac{8}{2} + 1) th$ or
$5^{th}$ 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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