Let `y = 1 + (a_(1))/(x - a_(1)) + (a_(2) x)/((x - a_(1))(x - a_(2))) + (a_(3) x^(2))/((x - a_(1))(x - a_(2))(x - a_(3))) + … (a_(n) x^(n - 1))/((x - a_(1))(x - a_(2))(x - a_(3))..(x - a_(n)))` Find `(dy)/(dx)`

( 1 + x + x^(2))^(n) = a_(0) + a_(1) x + a_(2) x^(2) + …+ a_(2n) x^(2n) , then a_(0) + a_(1) + a_(2) + a_(3) - a_(4) + … a_(2n) = .

Let (1 + x)^(n) = sum_(r=0)^(n) a_(r) x^(r) . Then ( a+ (a_(1))/(a_(0))) (1 + (a_(2))/(a_(1)))…(1 + (a_(n))/(a_(n-1))) is equal to

If (1 + x+ 2x^(2))^(20) = a_(0) + a_(1) x + a_(2) x^(2) + …+ a_(40) x^(40) . The value of a_(0) + a_(2) + a_(4) + …+ a_(38) is

If a_(1),a_(2),a_(3),a_(4),,……, a_(n-1),a_(n) " are distinct non-zero real numbers such that " (a_(1)^(2) + a_(2)^(2) + a_(3)^(2) + …..+ a_(n-1)^(2))x^2 + 2 (a_(1)a_(2) + a_(2)a_(3) + a_(3)a_(4) + ……+ a_(n-1) a_(n))x + (a_(2)^(2) +a_(3)^(2) + a_(4)^(2) +......+ a_(n)^(2)) le 0 " then " a_(1), a_(2), a_(3) ,....., a_(n-1), a_(n) are in

a1,a2,a3.....an are in H.P. (a_(1))/(a_(2)+a_(3)+ . . .+a_(n)),(a_(2))/(a_(1)+a_(3)+ . . . +a_(n)),(a_(3))/(a_(1)+a_(2)+a_(4)+ . . .+a_(n)), . . .,(a_(n))/(a_(1)+a_(2)+ . . . +a_(n-1)) are in

If (1 + x + x^(2) + x^(3))^(n)= a_(0) + a_(1)x + a_(2)x^(2) + a_(3) x^(3) +...+ a_(3n) x^(3n) , then the value of a_(0) + a_(4) +a_(8) + a_(12)+….. is

if (5x^(2) +2)/(x^(3)+x)=(A_(1))/(x)+(A_(2)x+A_(3))/(x^(2)+1), then (A_(1), A_(2), A_(3))=

If a_(1) = 2 and a_(n) - a_(n-1) = 2n (n ge 2) , find the value of a_(1) + a_(2) + a_(3)+…+a_(20) .

If (a_(2)a_(3))/(a_(1)a_(4))=(a_(2)+a_(3))/(a_(1)+a_(4))=3((a_(2)-a_(3))/(a_(1)-a_(4))) , then a_(1),a_(2),a_(3),a_(4) are in

Let (1 + x^(2))^(2) (1 + x)^(n) = a_(0) + a_(1) x + a_(2) x^(2) + … if a_(1),a_(2) " and " a_(3) are in A.P , the value of n is