The second, third and fourth terms in the binomial expansion $(x+a)^n$ are $240,720$ and $1080,$ respectively. Find $x, a$ and $n$
The second, third and fourth terms in the binomial expansion $(x+a)^n$ are $240,720$ and $1080,$ respectively. Find $x, a$ and $n$
Given that second term $T_{2}=240$
We have ${T_2} = {\,^n}{C_1}{x^{n - 1}} \cdot a$
So ${\,^n}{C_1}{x^{n - 1}} \cdot a = 240$ ..........$(1)$
Similarly ${\,^n}{C_2}{x^{n - 2}}{a^2} = 720$ ...........$(2)$
and $^{n} C_{x} x^{n-3} a^{3}=1080$ .............$(3)$
Dividing $(2)$ by $(1),$ we get
$\frac{{{\,^n}{C_2}{x^{n - 2}}{a^2}}}{{^n{C_1}{x^{n - 1}}a}} = \frac{{720}}{{240}}$ i.e., $\frac{(n-1) !}{(n-2) !} \cdot \frac{a}{x}=6$
or $\frac{a}{x}=\frac{6}{(n-1)}$ ...........$(4)$
Dividing $(3)$ by $(2),$ we have
$\frac{a}{x}=\frac{9}{2(n-2)}$ ...........$(5)$
From $(4)$ and $(5),$
$\frac{6}{n-1}=\frac{9}{2(n-2)}$ Thus, $n=5$
Hence, from $(1), 5 x^{4} a=240,$ and from $(4), \frac{a}{x}=\frac{3}{2}$
Solving these equations for $a$ and $x,$ we get $x=2$ and $a=3$
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