If `a_(m)` be the mth term of an AP, show that `a_(1)^(2)-a_(2)^(2)+a_(3)^(2)-a_(4)^(2)+"...."+a_(2n-1)^(2)-a_(2n)^(2)=(n)/((2n-1))(a_(1)^(2)-a_(2n)^(2))`.

If the sequence a_1, a_2, a_3,....... a_n ,dot forms an A.P., then prove that a_1^2-a_2^2+a_3^2-a_4^2+.......+ a_(2n-1)^2 - a_(2n)^2=n/(2n-1)(a_1^2-a_(2n)^2)

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

Differentiate |x|+a_(0)x^(n)+a_(1)x^(n-1)+a_(2)x^(n-2)+...+a_(n-1)x+a_(n)

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),...,A_(n),a_(1),a_(2),a_(3),...a_(n),a,b,c in R show that the roots of the equation (A_(1)^(2))/(x-a_(1))+(A_(2)^(2))/(x-a_(2))+(A_(3)^(2))/(x-a_(3))+…+(A_(n)^(2))/(x-a_(n)) =ab^(2)+c^(2) x+ac are real.

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

If 4a^(2)+9b^(2)+16c^(2)=2(3ab+6bc+4ca) , where a,b,c are non-zero real numbers, then a,b,c are in GP. Statement 2 If (a_(1)-a_(2))^(2)+(a_(2)-a_(3))^(2)+(a_(3)-a_(1))^(2)=0 , then a_(1)=a_(2)=a_(3),AA a_(1),a_(2),a_(3) in R .

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 a_(0),a_(1),a_(2) ,…, a_(2n) are the coefficients in the expansion of (1 + x + x^(2))^(n) in ascending power of x show that a_(0)^(2) - a_(1)^(2) + a_(2)^(2) -…+ a_(2n)^(2) = a_(n) .

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