Question

Three numbers are in $G.P$. whose sum is $70$ . If the extremes is each multiplied by $4$ and the mean by $5,$ they will be in A.P. Then the sum of numbers is

• A. 60
• B. 50
• C. 70
• D. None of these

Answer 1

Answer: (C)

Let the numbers be $a,ar,a{r}^2$.
Sum $=70$
$\Rightarrow a\left(1+r+r^2\right)=70(1)$
It is given that $4a,5ar,4a{r}^2$ are in $A.P$.
$\Rightarrow 2(5ar)=4a+4a{r}^2$
$\Rightarrow 5r=2+2r^2$
$\Rightarrow 2r^2-5r+2=0$
$\Rightarrow (2r-1)(r-2)=0$
$\Rightarrow r=2$ or $r=1/2$
Putting $r=2$ in $(1),$ we get $a=10$
Putting $r=1/2$ in (1), we get $a=40$.
Hence, the numbers are $10,20,40$ or $40,20,10$ .
Also sum of the numbers is $70$ .