Question

The second, third and fourth terms in the binomial expansion $(x+a{)}^n$ are $240,720$ and $1080$ respectively.
Based on the above information, which of the following is true?

• A. The value of $x$ is 3
• B. The value of a is 2
• C. The value of $n$ is 5
• D. None of these

Answer 1

Answer: (C)

Given that, second term $T_2=240$
We have, $T_2={}^n{C}_1{x}^{n-1}\cdot a$
So, ${}^n{C}_1{x}^{n-1}\cdot a=\mathrm{240.....}$(i)
Similarly, ${}^n{C}_2{x}^{n-2}\cdot a^2=\mathrm{720....}$(ii)
and ${}^n{C}_3{x}^{n-3}\cdot a^3=\mathrm{1080......}$(iii)
Dividing Eq. (ii) by Eq. (i), we get
$\frac{{}^n{C}_2{x}^{n-2}{a}^2}{{}^n{C}_1{x}^{n-1}a}=\frac{720}{240}\text{ i.e., }\frac{(n-1)!}{(n-2)!}\cdot \frac{a}{x}=6$
or $\frac{a}{x}=\frac{6}{(n-1)}.....$(iv)
Dividing Eq. (iii) by Eq. (ii), we get
$\frac{a}{x}=\frac{9}{2(n-2)}....$(v)
From Eqs. (iv) and (v), we get
$\frac{6}{n-1}=\frac{9}{2(n-2)}$
Thus, $n=5$
Hence, from Eq. (i), $5x^{4}a=240$, and from Eq. (iv), $\frac{a}{x}=\frac{3}{2}$
Solving these equations for $a$ and $x$, we get $x=2$ and $a=3$.