If `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)))` show that, `(dy)/(dx)=(y)/(x)[(a_(1))/(a_(1)-x)+(a_(2))/(a_(2)-x)+(a_(3))/(a_(3)-x)]`

If (x+1/x+1)^(6)=a_(0)+(a_(1)x+(b_(1))/(x))+(a_(2)x^(2)+(b_(2))/(x^(2)))+"...."+(a_(6)x^(6)+(b_(6))/(x^(6))) , then

If a_(1), a_(2), a_(3),…, a_(n) be in A.P. Show that, (1)/(a_(1)a_(2)) + (1)/(a_(2)a_(3)) +….+(1)/(a_(n-1)a_(n)) = (n-1)/(a_(1)a_(n))

If a_(1), a_(2), a_(3),…, a_(2k) are in A.P., prove that a_(1)^(2) - a_(2)^(2) + a_(3)^(2) - a_(4)^(2) +…+a_(2k-1)^(2) - a_(2k)^(2) = (k)/(2k-1)(a_(1)^(2) - a_(2k)^(2)) .

Prove that function y=(x-a_(1))^(2)+(x-a_(2))^(2)+...+(x-a_(n))^(2) is minimum when x=(1)/(n)(a_(1)+a_(2)+...+a_(n)) .

The asymptotes of the hyperbola (x^(2))/(a_(1)^(2))-(y^(2))/(b_(1)^(2))=1 and (x^(2))/(a_(2)^(2))-(y^(2))/(b_(2)^(2))=1 are perpendicular to each other. Then, (a) a_(1)/a_(2)=b_(1)/b_(2) (b) a_(1)a_(2)=b_(1)b_(2) (c) a_(1)a_(2)+b_(1)b_(2)=0 (d) a_(1)-a_(2)=b_(1)-b_(2)

If a_(1), a_(2),…,a_(n) are in G.P., then show that (1)/(a_(1)^(2) - a_(2)^(2)) + (1)/(a_(2)^(2) - a_(3)^(2))+...+ (1)/(a_(n-1)^(2) - a_(n)^(2)) = (r^(2))/((1-r^(2))^(2))[(1)/(a_(n)^(2))-(1)/(a_(1)^(2))]

If (1+x+x^(2))^(25)=a_(0)+a_(1)x+a_(2)x^(2)+...+a_(50).x^(50) , then a_(0)+a_(2)+a_(4)+....+a_(50) is

If the ellipse (x^(2))/(a_(1)^(2))+(y^(2))/(b_(1)^(2))=1(a_(1)^(2)gtb_(1)^(2)) has same eccentricity as that of the ellipse (x^(2))/(a_(2)^(2))+(y^(2))/(b_(2)^(2))=1(a_(2)^(2)gtb_(2)^(2)) prove that a_(1)b_(2) =a_(2)b_(1) .

If a_(1), a_(2) , a_(3),…,a _(n+1) are in A. P. then the value of (1)/(a _(1)a_(2))+(1)/(a_(2)a_(3))+(1)/(a_(3)a_(4))+...+(1)/(a_(n)a_(n+1)) is-

If a_(1) ge 0 for all t and a_(1), a_(2), a_(3),….,a_(n) are in A.P. then show that, (1)/(a_(1)a_(3)) + (1)/(a_(3)a_(5)) +...+ (1)/(a_(2n-1).a_(2n+1)) = (n)/(a_(1)a_(2n+1))