The terms of an infinitely decreasing G.P. in which all the terms are positive, the first term is 4, and the difference between the third and fifth terms is `32//81` , then `r=1//3` b. `r=2sqrt(2)//3` c. `S_oo=6` d. none of these

Options

A. `r=1//3`
B. `r= 2 sqrt(2)//3`
C. Sum of infinite terms is 6
D. none of these

Answer

A, B, C

Explanation

Given that `a=4,T_(3)-T_(5)=32//81`. Hence,
`a(r^(2)-r^(4))=32//81`.
o `r^(4)-r^(2)+8//81=0`
or `81r^(4)-81r^(2)+8=0`
or `(9r^(2)-8)(9r^(2)-1)=0`
`thereforer^(2)=8//9,1//9`
Therefore, the value of r is to be positive since all the terms are positive.
For r=1/3,
`S_(oo)=a/(1-r)=4/(1-1/3)=(4xx3)/2=6`
Similarly, we can find `S_(oo)` when `r=2sqrt2//3`.