Question

if 4th term in the expansion of ( ax+1/x)n is 5/2, then the values of a and n :

a) 1/2,6 b) 1,3

c) 1/2,3

Answer 1

Given -            (ax+1/x)ⁿ  
                         4th term in the expansion is 5/2
                        

Answer 2

Answer:

The correct option is a.

Step-by-step explanation:

The given expression is

$(ax+\frac{1}{x})^n$

The expansion of binomial is defined as

$(a+b)^n=^nC_0a^{n-0}b^0+^nC_1a^{n-1}b^1+...+^nC_{n-1}a^{1}b^{n-1}+^nC_na^{n-n}b^n$

The 4th term of the given expansion is 5/2.

$^nC_3(ax)^{n-3}(\frac{1}{x})^3=\frac{5}{2}$

$^nC_3(a)^{n-3}(x)^{n-3}(\frac{1}{x^3})=\frac{5}{2}$

$^nC_3(a)^{n-3}(x)^{n-6}=\frac{5}{2}$

On comparing both the sides, we get

$(x)^{n-6}=1$

It is possible if

$n-6=0$

The value of n is 6.

$^6C_3(a)^{6-3}=\frac{5}{2}$

$\frac{6!}{3!(6-3}!}(a)^{3}=\frac{5}{2}$

$20(a)^{3}=\frac{5}{2}$

Divide both sides by 20.

$(a)^{3}=\frac{1}{8}$

Cube root both sides.

$(a)=\frac{1}{2}$

The value of a is 1/2.

Therefore option a is correct.