If `A=sum_(r=7)^(2400)log_(7)((r+1)/(r))," "B=prod_(r=2)^(1023) log_(r)(t+1), " "C=sum_(r=2)^(2011) (1)/(log_(r)2+log_(r)3+.....+log_(r)2011)` then which of the following statements is/are true ?

Options

A. `3A+C=B`
B. `A+B+C=0`
C. `A+C=B`
D. `B-2A=4C`

Explanation

`underset(r=7)overset(2400)sumlog_(7)((r+1)/(r))=log_(7) ""(8)/(7)+log_(7)""(9)/(8)+log_(7)""(10)/(9)+....+log_(7)""(2401)/(2400)`
`=log_(7)""8/7xx9/8xx..........xx(2401)/(2400)log_(7)""((2401)/(7))=log_(7)343=3`
`B=underset(r=2)overset(1023)prodlog_(r)(r+1)=log_(2)3xxlog_(3)4xxlog_(4)5xx.....xxlog_(1023)1024=log_(2)1024=10`
`C=underset(r=2)overset(2011)sum(1)/(log_(r)2+log_(r)3+........+log_(r)2011)=underset(r=2)overset(2011)sum(1)/(log_(r)(2.3. .........2011))`
`=underset(r=2)overset(2011)sumlog_(((2.3....,2011)))r=log_((2.3.....2011))(2.3......2011)=1.`