Question

If the sum of first three terms of a G.P. is $\frac{39}{10}$ and their product is 1, then. its common ratio is $\frac{5}{2}or\frac{2}{5}$

• A. True
• B. False

Answer 1

The correct option is A

True

Let $\frac{a}{r},a$ and ar be the first three terms of the G.P.
Then, we have
$\frac{a}{r},a,ar=1\Rightarrow a^3=1\Rightarrow a=1$
Also, since the sum of these three terms is $\frac{39}{10}$,
we have
$\frac{a}{r},a,ar=\frac{39}{10}\Rightarrow a(\frac{1}{r}+1+r)=\frac{39}{10}$
$\Rightarrow \frac{1}{r}+1+r=\frac{39}{10}$ (As a = 1)
$\Rightarrow r+\frac{1}{r}=\frac{39}{10}-1=\frac{29}{10}$
$\Rightarrow \frac{r^2+1}{r}=\frac{29}{10}$
$\Rightarrow 10r^2+10=29r$
$\Rightarrow 10r^2-25r-4r+10=0$
$\Rightarrow 5r(2r-5)-2(2r-5)=0$
$\Rightarrow (2r-5)(5r-2)=0$
$\Rightarrow 2r-5=0$ or $5r-2=0$
$\Rightarrow r=\frac{5}{2}$ or $r=\frac{2}{5}$