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)`

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

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

If (x + a_(1)) (x + a_(2)) (x + a_(3)) …(x + a_(n)) = x^(n) + S_(1) x^(n-1) + S_(2) x^(n-2) + …+ S_(n) where , S_(1) = sum_(i=0)^(n) a_(i), S_(2) = (sumsum)_(1lei lt j le n) a_(i) a_(j) , S_(3) (sumsumsum)_(1le i ltk le n)a_(i) a_(j) a_(k) and so on . Coefficient of x^(7) in the expansion of (1 + x)^(2) (3 + x)^(3) (5 + x)^(4) is

If a_(1),a_(2),a_(3),"........",a_(n) are in AP with a_(1)=0 , prove that (a_(3))/(a_(2))+(a_(4))/(a_(3))+"......"+(a_(n))/(a_(n-1))-a_(2)((1)/(a_(2))+(1)/(a_(3))"+........"+(1)/(a_(n-2))) = (a_(n-1))/(a_(2))+(a_(2))/(a_(n-1)) .

If a_(1),a_(2),a_(3)"....." are in GP with first term a and common ratio r, then (a_(1)a_(2))/(a_(1)^(2)-a_(2)^(2))+(a_(2)a_(3))/(a_(2)^(2)-a_(3)^(2))+(a_(3)a_(4))/(a_(3)^(2)-a_(4)^(2))+"....."+(a_(n-1)a_(n))/(a_(n-1)^(2)-a_(n)^(2)) is equal to

If (1 - x + x^2)^n = a_0 + a_1 x + a_2x^2 + ..... + a_(2n)x^(2n) then a_0 + a_2 + a_4 + ... + a_(2n) equals

Let a_(1),a_(2),a_(3) be terms of an Ap. If (a_(1)+a_(2)+----+a_(p))/(a_(1)+a_(2)----a_(q))=(p^(2))/(q^(2)), p ne q, "then" (a_(6))/(a_(21)) is

the value of the determinant |{:((a_(1)-b_(1))^(2),,(a_(1)-b_(2))^(2),,(a_(1)-b_(3))^(2),,(a_(1)-b_(4))^(2)),((a_(2)-b_(1))^(2),,(a_(2)-b_(2))^(2) ,,(a_(2)-b_(3))^(2),,(a_(3)-b_(4))^(2)),((a_(3)-b_(1))^(2),,(a_(3)-b_(2))^(2),,(a_(3)-b_(3))^(2),,(a_(3)-b_(4))^(2)),((a_(4)-b_(1))^(2),,(a_(4)-b_(2))^(2),,(a_(4)-b_(3))^(2),,(a_(4)-b_(4))^(2)):}| is

Find the coefficient of x in the determinant |{:((1+x)^(a_(1)b_(1)),(1+x)^(a_(1)b_(2)),(1+x)^(a_(1)b_(3))),((1+x)^(a_(2)b_(1)),(1+x)^(a_(2)b_(2)),(1+x)^(a_(2)b_(3))),((1+x)^(a_(3)b_(1)),(1+x)^(a_(3)b_(2)),(1+x)^(a_(3)b_(3))):}|

If a_(1),a_(2),a_(3),a_(4),a_(5) are in HP, then a_(1)a_(2)+a_(2)a_(3)+a_(3)a_(4)+a_(4)a_(5) is equal to