UPSC NDA 2 2016 Mathematics Solved Paper

Let ‘a’ be the first term & ‘x’ be the common ratio.Also, suppose 27, 8 & 12 be the pth, qth & rth term of the GP.
∴axp−1=27
axq−1=8
and axr−1=12
Now,27×82=123
⇒axp−1x(axq−1)2=(axr−1)3
⇒xp−1⋅x2q−2=x2r−3
⇒p−1+2q−2=3r−3
⇒p+2q−3r=0...(1)
There are infinitely many solutions for the eq. (1).