Question

The sum of three numbers in GP is $70$. If the two extreme terms are multiplied by $4$, and the middle term by $5$, the resultants are in an AP. Find the numbers

• A. $10,25,35$
• B. $10,20,40$
• C. $15,30,35$
• D. $12,24,34$

Answer 1

Answer: (B)

$10,20,40$

Let the number be $a,ar,a{r}^2$.
According to question, $4a,5ar,4a{r}^2$ are in AP
$\Rightarrow 2\times 5ar=4a+4a{r}^2$
$\Rightarrow 4a{r}^2-10ar+4a=0$
$\Rightarrow 2a(2r^2-5r+2)=0$
$\Rightarrow 2r^2-5r+2=0$ ($\because a\ne 0$)
$\Rightarrow (2r-1)(r-2)=0$
$\Rightarrow r=\frac{1}{2},2$
Again given, $a+ar+a{r}^2=70$ ...(i)
When, $r=2$
from (i); $a+2a+4a=70\Rightarrow 7a=70\Rightarrow a=10$
Therefore, terms are $10,10\left(2\right),10\left(2^2\right)$i.e $10,20,40$
Option B is correct.