Question

The sum of first three terms of a $G.P.$ is $\frac{39}{10}$ and their product is $1.$ Find the common ratio and the terms.

Answer 1

Let the first three terms of $G.P.$ be $a,ar,a{r}^2$

Product is $a\left(ar\right)\left(a{r}^2\right)=1⟹ar=1⟹r=\frac{1}{a}$

Sum is $\frac{39}{10}⟹a+ar+a{r}^2=\frac{39}{10}⟹a+1+\frac{1}{a}=\frac{39}{10}⟹10(a^2+a+1)=39a$

$⟹10a^2-29a+10=0⟹a=\frac{5}{2},\frac{2}{5}$

The terms be $\frac{5}{2},1,\frac{2}{5}$ and the common ratio be $\frac{2}{5}$