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If `alpha, beta` be roots of the equation `375x ^(2) -25x-2=0 and s _(n) = alpha ^(n) +beta ^(n),` then `lim _( x to oo) lim _( x to oo) (sum _(r =1) ^(n ) S_(r))=`

A

`7/116`

B

`1/12`

C

`29/358`

D

none of these

Text Solution

Verified by Experts

The correct Answer is:
B
Since ` alpha, beta `are the roots of
`375x^2 - 25x -2 = 0 :. alpha + beta = 25/375 = 1/15`
and `alpha, beta - 2/375 :. lim_(nto oo)sum_(r=1)^n S_r = lim_(n tooo)sum_(r =1)^n (alpha^r +beta^r)`
`=(alpha + alpha^2 + alpha^3 +... ....oo) +(beta+beta^2 +beta^3 +.......oo)`
`=alpha/(1 -alpha) + beta/(1 -beta) = (alpha- alphabeta + beta - alphabeta)/((1-alpha)(1-beta))`
`=(alpha +beta- 2alphabeta)/(1 - (alpha+beta) + alphabeta) =(1/15 + 4/375)/(1 -1/15 - 2/375)`
`=(25 +4)/(375 - 25-2) =29/348 =1/12`
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