Let a,b,c are respectively the sums of the first n terms, the next n terms and the next n terms of a GP. Show that a,b,c are in GP.

Explanation

` a =S_n =(a(r^(n) -1))/( r-1) " "...(i) `
` b= S _ (2n ) -S_n =( a(r^(2n) -1))/( (r-1) ) -(a(r^(n) -1))/( (r-1) ) =(a(r^(n) -1))/( (r-1) )(r^(n))" "..(ii) `

` c= S_(3n ) -S_(2n ) =( a(r^(3n) -1))/( (r-1) ) =(a( r^(2n) - 1))/( (r-1)) `
`= ( a(r^(n) -1))/( (r-1) ) (r^(2n )+ r^(n) +1-r^(n) -1)= ( a( r^(n) -1))/( (r-1) ) (r^(n))^(2) `
From Eqs.(i) (ii) and (iii) ` b^(2) ` =ac,so a,b,c are in GP.