First slide

Binomial theorem for positive integral Index

Question

If the 4^th term in the expansion of (ax + 1/x)ⁿ is 5/2, then

Moderate

A

$a = \frac{1}{2}$
B

n = 8
C

$a = \frac{2}{3}$
D

n = 6

Solution

It is given that the fourth term in the expansion of ${(ax + \frac{1}{x})}^{n}$ is $\frac{5}{2}$ ; therefore,
$^{n} C_{3} (ax)^{n - 3} {(\frac{1}{x})}^{3} = \frac{5}{2} \Rightarrow^{n} C_{3} a^{n - 3} x^{n - 6} = \frac{5}{2}$ (1)
$\Rightarrow n = 6$ [ $∵$ R.H.S. is independent of x]
Putting n = 6 in (1), we get $^{6} C_{3} a^{3} = \frac{5}{2} or a^{3} = \frac{1}{8} or a = \frac{1}{2}$

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