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 numbers in G.P. are $\frac{a}{r},a,ar$

So,

$\frac{a}{r}\times a\times ar=216$

$\Rightarrow a^3=216$

$\Rightarrow a=6$

And also, given

$\frac{a}{r}+2,a+8,ar+6$ are in A.P.

$2(a+8)=(\frac{a}{r}+2)+(ar+6)$

$\Rightarrow 2(6+8)=\left(\frac{6+2r}{r}\right)+6r+6$

$\Rightarrow 28r=6+2r+6r^2+6r$

$\Rightarrow 6r^2-20r+6=0$

$\Rightarrow 6r^2-18r-2r+6=0$

$\Rightarrow 6r(r-3)-2(r-3)=0$

$\Rightarrow (r-3)(6r-2)=0$

$r=3,r=\frac{1}{3}$

Hence, putting values of and r, the required number are 18, 6, 2 or 2, 6 and 18.