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

(a_(1))/(a_(1)+a_(2))+(a_(3))/(a_(3)+a_(4))=(2a_(2))/(a_(2)+a_(3))

prove that (1)/((a-a_(1))^(2)),(1)/(a-a_(1)),(1)/(a_(1))(1)/((a-a_(2))^(2)),(1)/(a-a_(2)),(1)/(a_(2))(1)/((a-a_(3))^(2)),(1)/(a-a_(3)),(1)/(a_(3))]|=(-a^(2)(a_(1)-a_(2))(a_(2)-a_(3))(a_(3)-a_(1)))/(a_(1)a_(2)a_(3)(a-a_(1))^(2)(a-a_(2))^(2)(a-a_(3))^(2))

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

If a_(1),a_(2),a_(3)"....." are in GP with first term a and common rario 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

,1+a_(1),a_(2),a_(3)a_(1),1+a_(2),a_(3)a_(1),a_(2),1+a_(3)]|=0, then

if a_(1),a_(2),a+_(3)……,a_(12) are in A.P and Delta_(1)= |{:(a_(1)a_(5),,a_(1),,a_(2)),(a_(2)a_(6),,a_(2),,a_(3)),(a_(3)a_(7),,a_(3),,a_(4)):}| Delta_(3)= |{:(a_(2)b_(10),,a_(2),,a_(3)),(a_(3)a_(11),,a_(3),,a_(4)),(a_(3)a_(12),,a_(4),,a_(5)):}| then Delta_(2):Delta_(2)= "_____"

If a_(0),a_(1),a_(2),a_(3) are all the positive,then 4a_(0)x^(3)+3a_(1)x^(2)+2a_(2)x+a_(3)=0 has least one root in (-1,0) if ( a )a_(0)+a_(2)=a_(1)+a_(3) and 4a_(0)+2a_(2)>3a_(1)+a_(3)(b)4a_(0)+2a_(2)<3a_(1)+a_(3)(c)4a_(0)+2a_(2)=3a_(1)+a_(0) and 4a_(0)+a_(2)

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 a_(1),a_(2),...,.,a_(10) are positive numbers in arithmetic progression such that (1)/(a_(1)a_(2))+(1)/(a_(2)a_(3))+...*+(1)/(a_(9)a_(10))=(9)/(64) and (1)/(a_(1)a_(10))+(1)/(a_(2)a_(9))+......(1)/(a_(10)a_(1))=(1)/(10)((1)/(a_(1))+...+(1)/(a_(10))) then the value of (2)/(5)((a_(1))/(a_(10))+(a_(10))/(a_(1))) is