व्यंजक `2(1+(1)/(omega))(1+(1)/(omega^(2))) + 3(2 + (1)/(omega))(2+(1)/(omega^(2))) + 4(3 + (1)/(omega))(3 + (1)/(omega^(2))) + … + (n + 1) (n + (1)/(omega))(n + (1)/(omega^(2)))` का मान है

Options

A. `(n(n^(2) + 2))/(3)`
B. `(n(n^(2) -2))/(3)`
C. `(n^(2)(n + 1)^(2) + 4n)/(4)`
D. इनमे से कोई नहीं

Answer

C

Explanation

दिए गए व्यंजक में rवां पद,
`T_(r) = (r +1) (r + (1)/(omega))(r + (1)/(omega^(2)))`
` = r^(3) + r^(2) ((1)/(omega^(2)) + (1)/(omega) + 1) + r (1 + (1)/(omega) + (1)/(omega^(2))) +1`
` = r^(3) +r^(2) ((omega + omega^(2) + 1)/(omega^(2))) + r ((omega + omega^(2) + 1)/(omega^(2))) + 1`
` = r^(3) + r^(2) * 0 + r * 0 + 1 = r^(3) + 1`
`therefore S_(n) = sum_(r = 1)^(n) (r^(3) + 1) = {(n(n+1))/(2)}^(2) + n`
` = (n^(2)(n +1)^(2))/(4) + n = (n^(2)(n +1)^(2) + 4n)/(4)`