Question

Find four number forming a $G.P.$ in which the third term is greater than the first by $9$, and the second term is greater than the fourth by $18$. Write the largest of the four numbers.

Answer 1

Let the four numbers in $GP$ be a, ar, $a{r}^2$ ,$a{r}^3$

Now, given that

$a{r}^2=a+9$

$\Rightarrow a{r}^2-a=9$

$\Rightarrow a\left(r^2-1\right)=9$

$\Rightarrow -a\left(1-r^2\right)=9$

$\Rightarrow 1-r^2=-\frac{9}{a}-\left(1\right)$

$ar=a{r}^3+18$

$\Rightarrow ar-a{r}^3=18$

$\Rightarrow ar\left(1-r^2\right)=18$

$\Rightarrow ar\times \frac{-9}{a}=18$

$\Rightarrow r=\frac{-18}{9}$

$\Rightarrow r=-2$

$puttingr=-2in$

$a{r}^2=a+9$

$\Rightarrow a{\left(-2\right)}^2=a+9$

$\Rightarrow 4a=a+9$

$\Rightarrow 3a=9$

$\Rightarrow a=3$

$3,-6,12,-36$