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