If a,b,c are in geometric progression and a,2b,3c are in arithmetic progression, then what is the common ratio r such that `0ltrlt1` ?

Options

A. `(1)/(3)`
B. `(1)/(2)`
C. `(1)/(4)`
D. `(1)/(8)`

Answer

A

Explanation

Given that a, b, c are in GP.
Let r be common ratio of GP.
So, a=a, b=ar and`c=ar^(2)" "…(i)`
Also, given that a, 2b, 3c are in AP.
`implies" "2b=(a+3c)/(2)`
`implies" "4b=a+3c`
From Eq. (1)
`4ar=a+3ar^(2)`
`implies" "3ar^(2)-4ar+a=0`
`implies" "3r^(2)-4r+1=0`
`implies" "3r^(2)-3r-r+1=0`
`implies" "3r(r-1)-1(r-1)=0`
`implies" "(r-1)(3r-1)=0`
`implies" "r=1" or "r=(1)/(3),r=(1)/(3)` is in the choice.