how to solve this question? Please tell the solution of this question.I am not understanding how to solve this.

Let the first term be a and common difference = d. 

The given relation yields a+5d = 3......1

 

Since a is not an integer, d cannot be an integer, but its rational

 

Let the fractional part of a be f₀ and that of d be f₁.

 

Since every alternate term is an integer, this means 2f₁ is an integer. Hence f₁= ½ 

 

From eqn 1, its now clear that f₀ = ½ 

 

So let a = m+ ½ and d = n + ½ with m>0

 

So from Eqn 1, we get m+5n = 0. So, possible pairs of m, n are (5,-1), (10,-2), (15,-3) etc. and correspondingly (a,d) can be (11/2, -1/2), (21/2, -3/2), (31/2, -5/2) and so on.

 

So possible choices for 3^rd term are 4,6,8...

 

Of the given alternatives, Option (D) i.e. 6 is the answer.