Question

The sum of three numbers in GP with common ratio greater than 1 is 84 . If the first, second and third number are multiplied by 16, 12 and 5 respectively then the resulting terms are in AP Find the numbers of AP

• A. 16
• B. 24
• C. 28
• D. 20

Answer 1

Answer: (A), (B), (D)

Let the numbers are $\frac{a}{r},a,ar$
Given that
$\frac{a}{r}+a+ar=84$
$a+ar+a{r}^2=84r$ ...(i)
Also $\frac{16a}{r},12a$, and $5ar$ are in AP
$\Rightarrow 2\times 12a=\frac{16a}{r}+5ar$
$\Rightarrow 24=\frac{16}{r}+5r$
$\Rightarrow 24r=16+5r^2$
$\Rightarrow 5r^2-24r+16=0$
$\Rightarrow 5r^2-20r-4r+16=0$
$\Rightarrow 5r(r-4)-4(r-4)=0$
$\Rightarrow (r-4)(5r-4)=0$
$\Rightarrow r=4,\frac{4}{5}\Rightarrow r=4$
$[\because r>1]$
So, put $r=4$ in equation (i)
$a+4a+16a=84\times 4$
$\Rightarrow 21a=84\times 4$
$\Rightarrow a=\frac{84\times 4}{21}=16$
Hence, the required numbers are 16,20 , and 24.