Question

Find three number in $G.P.$ such that their sum is $35$ and their product is $1000$.

Answer 1

To find three no.in $G.P.$ such that sum is $35$ and product is $1000$

Let $a,ar,a{r}^2$ are three no.in $G.P$

given $a+ar+a{r}^2=35\Rightarrow a(1+r+r^2)=35$ - $\left(1\right)$

$a.ar.a{r}^2=1000$

$a^3{r}^3=1000$

$a^3=\left(\frac{1000}{r^3}\right)$

$a=\frac{{\left(1000\right)}^{1/2}}{r}=\frac{10}{r}$

Put the value of a in equation $\left(1\right)$

Now $\frac{10}{r}(1+r+r^2)=35$

$\frac{1}{r}+1+r=\frac{35}{10}$

$r+\frac{1}{r}=\frac{25}{10}=\frac{5}{2}$

$2(r^2+1)=5x$

$\Rightarrow 2r^2-5r+2=0$

$\Rightarrow (2r-1)(r-2)=0$

$\Rightarrow r=\frac{1}{2}$ or $r=2$

If $r=\frac{1}{2},a=20$ then $a(1+r+r^2)=35$ and $a^3{r}^3=1000$

If $r=2$, then $a=5$ then $a(1+r+r^2)=35,a^3{r}^3=1000$

So there three no. are

If $a=20$ and $r=1/2$ then $(20,20\times \frac{1}{2},20\times \frac{1}{4})$

$\Rightarrow (20,10,5)$

If $a=5$ adn $r=2$ then $(5,10,20)$