The second, third and fourth terms in the binomial expansion (x+a) n are 240,720 and1080 respectively. Find x and n.

bySCIENCE AND TECHNOLOGY -September 01, 2021

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

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)

$\frac{( 2 )}{( 3 )} \Rightarrow \frac{( n ) ( n - 1 ) ( 3 ) ( 2 )}{2 ( n ) ( n - 1 ) ( n - 2 )} \frac{a}{x} = \frac{7 2 0}{1 0 8 0}$

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

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

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

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

$\Rightarrow n = 5$ ∴ax=23
$a = \frac{2 x}{3};$ Now equation (1) $\Rightarrow (n) (n) (\frac{2 x}{3})^{4} = 240$
$\Rightarrow x^{5} = \frac{2 4 0}{5} (\frac{3}{2})^{4} \Rightarrow x = 3$

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