Let `a,b,c,d` be positive real numbers with `altbltcltd`. Given that `a,b,c,d` are the first four terms of an AP and `a,b,d` are in GP. The value of `(ad)/(bc)` is `(p)/(q)`, where `p` and `q` are prime numbers, then the value of `q` is _____

Explanation

a,b,c,d are positive real numbers with
`altbltcltd" " "………(A)"`
According to the question, a,b,c,d are in AP.
`implies b=a+alpha,c=a+2alpha` and `d=a+3alpha" " "………..(i)"`
`alpha` be the common difference
and a,b,c,d are in GP.
`implies b^(2)=ad" " ".........(ii)"`
From Eqs. (i) and (ii), we get
`(a+alpha)^(2)=a(a+3alpha)`
`impliesa^(2)+alpha^(2)+2aalpha=a^(2)+3aalpha`
`implies alpha^(2)=a alpha`
`implies a(alpha-a)=0`
`implies alpha=0 " or "alpha=a`
`ane 0 " by "(A), " so "alpha=a`
From Eq. (i), `b=2a,c=3a " and "d=4a`
`(ad)/(bc)=(a*4a)/(2a*3a)=(2)/(3)=((p)/(q))`
where, p and q are prime numbers.
So,`q=3`.