The product of three numbers in G.P. is 125 and the sum of their products taken in pairs is 871 / 2. Find them.

Home » The product of three numbers in G.P. is 125 and the sum of their products taken in pairs is 871 / 2. Find them.

The product of three numbers in G.P. is 125 and the sum of their products taken in pairs is 871 / 2. Find them.

Solution:

Let the three numbers be $a/r$ , $a$ , $ar$
According to the question
$\mathrm{a} / \mathrm{r} \times \mathrm{a} \times \mathrm{ar}=125 \ldots$ eq.(1)
From eq.(1) we obtain,
$\begin{array}{l} a^{3}=125 \\ a=5 \end{array}$
$\mathrm{a} / \mathrm{r} \times \mathrm{a}+\mathrm{a} \times \mathrm{ar}+\mathrm{ar} \times \mathrm{a} / \mathrm{r}=87 {\frac{1}{2}}$
$a / r \times a+a \times a r+a r \times a / r=195 / 2$
$a^{2} / r+a^{2} r+a^{2}=195 / 2$
$a^{2}(1 / r+r+1)=195 / 2$
On substituting $a=5$ in above equation we obtain,
$\begin{array}{l} 5^{2}\left[\left(1+r^{2}+r\right) / r\right]=195 / 2 \\ 1+r^{2}+r=(195 r / 2 \times 25) \\ 2\left(1+r^{2}+r\right)=39 r / 5 \\ 10+10 r^{2}+10 r=39 r \\ 10 r^{2}-29 r+10=0 \\ 10 r^{2}-25 r-4 r+10=0 \\ 5 r(2 r-5)-2(2 r-5)=0 \\ r=5 / 2,2 / 5 \end{array}$
Therefore the G.P is $10,5,5 / 2$ or $5 / 2,5,10$
As a result, the three numbers are $10,5,5 / 2$ or $5 / 2,5,10$