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, ${ }^n C_1 x^{n-1} \cdot a=240 .....$(i)
Similarly, ${ }^n C_2 x^{n-2} \cdot a^2=720 ....$(ii)
and ${ }^n C_3 x^{n-3} \cdot a^3=1080......$(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), $5 x^4 a=240$, and from Eq. (iv), $\frac{a}{x}=\frac{3}{2}$
Solving these equations for $a$ and $x$, we get $x=2$ and $a=3$.