If `A=1+r^a+r^(2a)+...oo=a` and `B=1+r^b+r^(2b)+...oo=b` then `a/b` is equal to

Options

A. `log_(1-B)(1-A)`
B. `log_(((B-1)/(B)))((A-1)/(A))`
C. `log_(B)A`
D. none of these

Answer

B

Explanation

We have,
`A=(1)/(1-r^(a)),B=(1)/(1-r^(b))`
`rArr" "r=((A-1)/(A))^((1)/(a))=((B-1)/(B))^((1)/(b))`
`rArr" "((A-1)/(A))^(b)=((B-1)/(B))^(a)`
`rArr" "alog((B-1)/(B))=blog((A-1)/(A))rArr(a)/(b)=log_(((B-1)/(B)))((A-1)/(A))`