First slide

Binomial theorem for positive integral Index

Question

The sum of the real values of x for which the middle term in the binomial expansion of ${(\frac{x^{3}}{3} + \frac{3}{x})}^{8}$ equals 5670 is

Moderate

A

4
B

0
C

6
D

8

Solution

ln expansion of ${(\frac{x^{3}}{3} + \frac{3}{x})}^{8}$ the middle term is $T_{4 + 1}$
$[∵ here, n = 8$ which is even, therefore middle term $= (\frac{n + 2}{2})$ th term $]$
$\begin{array}{l} ∴ 5670 =^{8} C_{4} {(\frac{x^{3}}{3})}^{4} {(\frac{3}{x})}^{4} = \frac{8 \cdot 7 \cdot 6 \cdot 5}{1 \cdot 2 \cdot 3 \cdot 4} x^{8} \\ [∵ T_{r + 1} =^{8} C_{r} {(\frac{x^{3}}{3})}^{8 - r} {(\frac{3}{x})}^{r}] \\ \Rightarrow x^{8} = 3^{4} \Rightarrow x = \pm \sqrt{3} \end{array}$
So, sum of all values of x i.e $+ \sqrt{3} and - \sqrt{3} = 0$

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