If ` alpha and beta` are the roots of the equation `375x^(2) - 25x -2 = 0," then " sum_(r=1)^(n) alpha^(r) + sum_(r=1)^(n) beta^(r)` is equal to

Options

A. `21/346`
B. ` 29/358`
C. ` 1/12`
D. ` 7/116`

Answer

C

Explanation

Given `alpha and beta ` are roots of quadratic equation ` 375x^(2) - 25x - 2 = 0 `
`:." " alpha + beta = (25)/(375) = 1/15` ….(i)
and `alpha beta =- 2/(375)` …(ii)
Now, ` underset( n to infty) limunderset(r=1)overset(n) sum alpha^(r) + underset( n to infty) lim underset( r = 1) overset( n) sum beta^(r) `
` = (alpha+alpha^(2) + alpha^(3) + ...+` upto infinite terms)+ (`beta+beta^(2)+beta^(3)+...+` upto infinite terms)
` = alpha/(1-alpha)+ beta/(1-beta)" "[:' S_(infty) = a/(1-r) ` for GP]
` = (alpha(1-beta)+beta(1-alpha))/((1-alpha)(1-beta))=(alpha - alpha beta + beta - alpha beta)/(1-alpha-beta + alpha beta)= ((alpha+beta)-2alpha beta)/(1-(alpha+beta)+alpha beta)`
On substituting the value `alpha + beta = 1/15 and alpha beta = (-2)/(375)` from Eqs. (i) and (ii) respectively,
we get
`= (1/15+4/375)/(1-1/15-2/375) = 29/(375-25-2)=29/348=1/12`