Question

Sum of three numbers in $GP$ is $13$ and their product is $27$. Find the numbers.

Answer 1

Sum of three numbers of $GP$ is $13$

$a+ar+a{r}^2=13..........\left(1\right)$

Product of three,

$a\times ar\times a{r}^3=27.............\left(2\right)$

$=>a^3{r}^3=27$

$=>\left(ar\right)=3$ $\because r=\frac{3}{a}$

$=>(a+ar+a{r}^2)=13$

$=>(a+3+a{r}^2)=13$

$=>a(1+r^2)=10$

$=>a(1+\frac{9}{a^2})=10$

$=>a+\frac{9}{a}=10$

$=>a^2+9-10a=0$

$=>a^2-10a+9=0$

$=>a^2-9a-a+9=0$

$=>a(a-9)-1(a-9)=0$

$a=9,a=1$

Series can be $\left(1\right)$ $1,3,9,...............$

$\left(2\right)$ $9,3,1,...........$