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))=`

Options

A. `(7)/(116)`
B. `(1)/(12)`
C. `(29)/(358)`
D. none of these

Answer

B

Explanation

Since `alpha, beta` are the roots of
`375x^(2)-25x-2=0 therefore alpha+beta=(25)/(375)=(1)/(15)`
and `alpha beta= -(2)/(375) therefore lim_(n to oo) sum_(r=1)^(n)S_(r)=lim_(n to oo) 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-alpha beta+beta-alpha beta)/((1-alpha)(1-beta))`
`=(alpha+beta-2alpha beta)/(1-(alpha+beta)+alpha beta)= ((1)/(15)+(4)/(375))/(1-(1)/(15)-(2)/(375))`
`=(25+4)/(375-25-2)=(29)/(348)=(1)/(12)`.