The second, third and fourth terms in the binomial expansion ${(x + a)}^{n}$ are $240, 720$ and $1080$ respectively. Find $x$ and $n$ .

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Solution

$^{n} C_{1} x a^{n - 1} = 240$ .......(1)
$^{n} C_{2} x^{2} a^{n - 2} = 720$ .........(2)
$^{n} C_{2} x^{3} a^{n - 3} = 1080$ ........(3)

$\frac{(1)}{(2)} \Rightarrow \frac{2 n}{(n) (n - 1)} \frac{a}{x} = \frac{1}{3}$

$6 = \frac{(n) (n - 1)}{a}$ .......(4)

$\frac{(2)}{(3)} \Rightarrow \frac{(n) (n - 1) (3) (2)}{2 (n) (n - 1) (n - 2)} \frac{a}{x} = \frac{720}{1080}$

$\Rightarrow \frac{3 a}{(n - 2) x} = \frac{2}{3} \Rightarrow \frac{9}{2} \frac{(x) (x - 2)}{a}$ ........(5)

$\frac{1}{3} \Rightarrow \frac{n a^{2} (3) (2)}{(n) (n - 1) (n - 2) x^{2}} = \frac{2}{9}$

$\Rightarrow \frac{3 a^{2}}{(n - 1) (n - 2) x^{2}} = \frac{1}{9}$

$\frac{4}{5} \Rightarrow \frac{2 \times 6}{9} = \frac{n - 1}{n - 2} = \frac{4}{3}$

$\Rightarrow n = 5 ∴ \frac{x}{a} = \frac{3}{2}$
$a = \frac{2 x}{3};$ Now equation (1) $\Rightarrow (n) (n) {(\frac{2 x}{3})}^{4} = 240$
$\Rightarrow x^{5} = \frac{240}{5} {(\frac{3}{2})}^{4} \Rightarrow x = 3$

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