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.
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.
Let $\frac{a}{r}, a,$ ar be the first three terms of the $G.P.$
$\frac{a}{r}+a+a r=\frac{39}{10}$ ..........$(1)$
$\left(\frac{a}{r}\right)(a)(a r)=1$ .........$(2)$
From $(2),$ we Obtain $a^{3}=1$
$\Rightarrow a=1$ (Considering real roots only)
Substituting $a=1$ in equation $(1),$ we obtain
$\frac{1}{r}+1+r=\frac{39}{10}$
$\Rightarrow 1+r+r^{2}=\frac{39}{10} r$
$\Rightarrow 10+10 r+10 r^{2}-39 r=0$
$\Rightarrow 10 r^{2}-29 r+10=0$
$\Rightarrow 10 r^{2}-25 r-4 r+10=0$
$\Rightarrow 5 r(2 r-5)-2(2 r-5)=0$
$\Rightarrow(5 r-2)(2 r-5)=0$
$\Rightarrow r=\frac{2}{5}$ or $\frac{5}{2}$
Thus, the three terms of $G.P.$ are $\frac{5}{2}, 1$ and $\frac{2}{5}$
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