If `a_1, a_2, a_3 ......a_n (n>= 2) ` are read and `(n-1) a_1^2 -2na_2 < 0` then prove that at least two roots of the equation ` x^n+a_1 x^(n-1) +a_2 x^(n-2) +......+a_n = 0 `are imaginary.

If a_1, a_2, a_3 ......a_n (n>= 2) are real and (n-1) a_1^2 -2na_2 < 0 then prove that at least two roots of the equation x^n+a_1 x^(n-1) +a_2 x^(n-2) +......+a_n = 0 are imaginary.

If a_1, a_2, a_3 ......a_n (n>= 2) are real and (n-1) a_1^2 -2na_2 < 0 then prove that at least two roots of the equation x^n+a_1 x^(n-1) +a_2 x^(n-2) +......+a_n = 0 are imaginary.

If a_1, a_2, a_3 ......a_n (n>= 2) are real and (n-1) a_1^2 -2na_2 < 0 then prove that at least two roots of the equation x^n+a_1 x^(n-1) +a_2 x^(n-2) +......+a_n = 0 are imaginary.

If a_1. a_2 ....... a_n are positive and (n - 1) s = a_1 + a_2 +.....+a_n then prove that (a_1 + a_2 +....+a_n)^n ge (n^2 - n)^n (s - a_1) (s - a_2)........(s - a_n)

If a_1, a_2, a_3,.....a_n are in H.P. and a_1 a_2+a_2 a_3+a_3 a_4+.......a_(n-1) a_n=ka_1 a_n , then k is equal to

If a_1, a_2, a_3,.....a_n are in H.P. and a_1 a_2+a_2 a_3+a_3 a_4+.......a_(n-1) a_n=ka_1 a_n , then k is equal to

If a_1, a_2, a_3,.....a_n are in H.P. and a_1 a_2+a_2 a_3+a_3 a_4+.......a_(n-1) a_n=ka_1 a_n , then k is equal to

If a_1, a_2, a_3 ….. a_n are in AP, then prove that frac(1)(a_1 a_2)+frac(1)(a_2 a_3)+frac(1)(a_3 a_4)+.....+frac(1)(a_(n-1) a_n)=frac(n-1)(a_1 a_n)

Let the sequence a_1 , a_2 , a_3 ......... a_n form an A.P. then a_1^2 - a_2^2 + a_3^2 - a_4^2 +.....+ a_(2n-1)^2 - a_(2n)^2 is equal to:- (1) n/(2n-1)(a_1^2-a_(2n)^2) (2) (2n)/(n-1)(a_(2n)^2-a_1^2) (3) n/(n+1)(a_1^2+a_(2n)^2) (4)none of these

Let the sequence a_1 , a_2 , a_3 ......... a_n form an A.P. then a_1^2 - a_2^2 + a_3^2 - a_4^2 +.....+ a_(2n-1)^2 - a_(2n)^2 is equal to:- (1) n/(2n-1)(a_1^2-a_(2n)^2) (2) (2n)/(n-1)(a_(2n)^2-a_1^2) (3) n/(n+1)(a_1^2+a_(2n)^2) (4)none of these