Question

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.

Answer 1

Let the terms of the given G.P. be $\frac{a}{r},a\mathrm{and}ar$.
∴ Product = 216
$$\begin{gathered} \Rightarrow a^3=216 \\ \Rightarrow a=6 \end{gathered}$$
It is given that $\frac{a}{r}+2,a+8\mathrm{and}ar+6$ are in A.P.
$$\begin{gathered} \therefore 2\left(a+8\right)=\frac{a}{r}+2+ar+6 \\ \mathrm{Putting}a=6,\mathrm{we}\mathrm{get} \\ \Rightarrow 28=\frac{6}{r}+2+6r+6 \\ \Rightarrow 28r=6r^2+8r+6 \\ \Rightarrow 6r^2-20r+6=0 \\ \Rightarrow \left(6r-2\right)\left(r-3\right)=0 \\ \Rightarrow r=\frac{1}{3},3 \\ \mathrm{Hence},\mathrm{putting}\mathrm{the}\mathrm{values}\mathrm{of}a\mathrm{and}r,\mathrm{the}\mathrm{required}\mathrm{numbers}\mathrm{are}18,6,2\mathrm{or}2,6\mathrm{and}\mathit{}18. \end{gathered}$$