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Let T(r) be the r^(th) term of a sequenc...

Let `T_(r)` be the `r^(th)` term of a sequence, for r=1,2,3,......... If `3T_(r+1)=T_(r)` and `T_(7)=(1)/(243)` then the value of `sum_(r=1)^(oo)(T_(r).T_(r+1))` is :

A

`(9)/(2)`

B

`(27)/(8)`

C

`(81)/(8)`

D

`(81)/(4)`

Text Solution

Verified by Experts

The correct Answer is:
B
`(T_(r+1))/(T_(r))=(1)/(3)`
`impliesT_(1),T_(2),T_(3)...................G.P.`
`T_(7)=(1)/(243)`
`ar^(6)=(1)/(243)`
`(a)/(3^(6))=(1)/(3^(5))impliesa=3`
Sum of infinite series `=T_(1)T_(2)+T_(2)T_(3)+T_(3)T_(4)............`
`(T_(1)T_(2))/(1-r^(2))=(3xx3(1)/(3))/(1-(1)/(9))=(27)/(8)`
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