The sum of the first three erms of a GP is 7/9 and their product is ` -8//27` Find the common ratio of the series

Options

A. `r = -2 2//3 or -3//2`
B. ` r =-2//3 or 3//2`
C. `r=2//3 or -3//2`
D. ` r = 2//3 or 3//2`

Answer

a

Explanation

Let first three terms of a GP be a,ar,`ar^(2)`
Then, `"sum" 7/9`
` rArr a+ar+ar^(2) =7/9`
` rArr a(1+r_r^(2)) =7/9" "` …(i)
and `" product" = -8/27`
` rArr a* ar* ar^(2) -8/27 rArr a^(3)r^(3) = -8/27 = ((-2)/3)^(3)`
On comparing the powers , we get
` "ar"=(-2)/3 " "` ...(ii)
On dividing Eq.(i) by Eq.(ii) , we get
` (1+r+r^(2))/r = (7/9)/((-2)/3)`
` rArr (1+r+r^(2))/r = 7/9 xx3/2 = (-7)/6`
`rArr 1+ r+r^(2) = (-7)/6 r`
`rArr r^(2) +(1+7/6)r +1=0`
` r^(2) +13/6 r+1 =0`
`rArr (r+3/2)(r+2/3)=0`