If `x^2+x+1=0` then find the value of `sum_(r=1)^27 (x^r+1/(x^r))^2`

If x^(2)-x+1=0, then the value of sum_(r=1)^(2010)(x^(r)-(1)/(x^(r)))^(3) is equal to

If n in N such that is not a multiple of 3 and (1+x+x^(2))^(n) = sum_(r=0)^(2n) a_(r ). X^(r ) , then find the value of sum_(r=0)^(n) (-1)^(r ).a_(r).""^(n)C_(r ) .

If n in N such that is not a multiple of 3 and (1+x+x^(2))^(n) = sum_(r=0)^(2n) a_(r ). X^(r ) , then find the value of sum_(r=0)^(n) (-1)^(r ).a_(r).""^(n)C_(r ) .

If f(x)=(a^x)/(a^x+sqrt(a ,)),(a >0), then find the value of sum_(r=1)^(2n-1)2f(r/(2n))

If f(x)=(a^x)/(a^x+sqrt(a ,)),(a >0), then find the value of sum_(r=1)^(2n1)2f(r/(2n))

If alpha and beta are the roots of the quadratic equation 4x ^(2) + 2x -1=0 then the value of sum _(r =1) ^(oo) (a ^(r ) + beta ^(r )) is :

If alpha and beta are the roots of the quadratic equation 4x ^(2) + 2x -1=0 then the value of sum _(r =1) ^(oo) (a ^(r ) + beta ^(r )) is :

If f(x)=(a^(x))/(a^(x)+sqrt(a)),(a>0), then find the value of sum_(r=1)^(2n1)2f((r)/(2n))

If n is a positive integer and C_r = ""^nC_r then find the value of sum_(x = 1)^n r^2 ((C_r)/(C_(r - 1)) )

Find sum_(r=1)^10 (3x 2^r )