The product of three numbers in G.P. is 216. If 2, 8, 6 be added to them, the results are in A.P. Find the numbers.
Let the numbers in G.P. are ar,a,ar
So,
ar×a×ar=216
⇒a3=216
⇒a=6
And also, given
ar+2,a+8,ar+6 are in A.P.
2(a+8)=(ar+2)+(ar+6)
⇒2(6+8)=(6+2rr)+6r+6
⇒28r=6+2r+6r2+6r
⇒6r2−20r+6=0
⇒6r2−18r−2r+6=0
⇒6r(r−3)−2(r−3)=0
⇒(r−3)(6r−2)=0
r=3,r=13
Hence, putting values of and r, the required number are 18, 6, 2 or 2, 6 and 18.