Question

The product of three consecutive terms of a G.P. is $216$ . If $3$ is added to each of the first and the second of these terms, the three form an A.P. Find the sum of the original three terms.

Answer 1

Answer: 21

Let the three terms in G.P. be $\frac{a}{r},a$ and $ar$.
Product of three terms $=216$
$\Rightarrow \frac{a}{r}\cdot a\cdot ar=216$
$\Rightarrow a^3=6^3$
$\Rightarrow a=6$
Three terms in A.P. are $\frac{6}{r}+3,6+3$ and $6r$
i.e. $\frac{6}{r}+3,9$ and $6r$
$\Rightarrow 18=\frac{6}{r}+3+6r$
$\Rightarrow 2r^2-5r+2=0$
$\Rightarrow (r-2)(2r-1)$
$\Rightarrow r=2$
or $r=\frac{1}{2}$
The three terms in G.P. are $3,6,12$ or $12,6,3$.
$\Rightarrow$ Required sum $=3+6+12=21$