First slide

Binomial theorem for positive integral Index

Question

If $a > 0$ and coefficient of $x^{2}, x^{3}, x^{4}$ in the expansion of ${(1 + \frac{x}{a})}^{6}$ are in $A . P .,$ then a equals

Moderate

A

$\frac{(4 + \sqrt{7})}{3}$
B

$\frac{(4 + \sqrt{3})}{3}$
C

$2 - \sqrt{3}$
D

$2 + \sqrt{3}$

Solution

Coefficient of $x^{r}$ in the expansion of
$(1 + x / a)^{6} =^{6} C_{r} (1 / a)^{r}$
According to given condition $^{6} C_{2} (1 / a)^{2},^{6} C_{3} (1 / a)^{3},^{6} C_{4} (1 / a)^{4}$
are in A.P.
$\begin{array}{l} ∴ 2 (^{6} C_{3}) {(\frac{1}{a})}^{3} =^{6} C_{2} {(\frac{1}{a})}^{2} +^{6} C_{4} {(\frac{1}{a})}^{4} \\ \Rightarrow 2 (20) a = 15 (a^{2} + 1) \\ \Rightarrow 3 a^{2} - 8 a + 3 = 0 \\ \Rightarrow a = \frac{(4 + \sqrt{7})}{3} as a > 0 . \end{array}$

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