Q. $If\ the\ fourth\ term\ in\ the\ expansion\ of\ \left( ax + \frac{1}{x} \right)^{n}\ is\frac{5}{2}\ ,then$
A) $a = 1/2\ and\ n = 6$
B) $a = 1/3\ and\ n = 5$
C) $a = 2\ and\ n = 3$
D) $a = 1/4\ and\ n = 1$
Explanation
(a)
$T_{4} = \ ^{n}C_{3}(ax)^{n - 3}\left( \frac{1}{x} \right)^{3} = \frac{5}{2}\ \ \ \ \ \ \ \ \ \lbrack given\rbrack$
$\Longrightarrow \ \ ^{n}C_{3}a^{n - 3}x^{n - 6} = \frac{5}{2}$
$\Longrightarrow n - 6 = 0$ [∵ RHS is independent of $x$]
$\Longrightarrow n = 6$
On putting $n = 6$ in Eq. (i), we get
$\ ^{6}C_{3}a^{3} = \frac{5}{2} \Longrightarrow a^{3} = \frac{1}{8} \Longrightarrow a = \frac{1}{2}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.