" (iv) "|[1+a_(1),a_(2),a_(3)],[a_(1),1+a_(2),a_(3)],[a_(1),a_(2),1+a_(3)]|=1+a_(1)+a_(2)+a_(3).

,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)= "_____"

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

If a_(0),a_(1),a_(3),a_(4)......... are in AP with common difference d where d!=0,1,-1, then the det[[a_(1)a_(2),a_(1),a_(0)a_(2)a_(3),a_(2),a_(1)a_(3)a_(4),a_(3),a_(2)]] is

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 eqiual to

Suppose a_(1),a_(2),a_(3) are in A.P. and b_(1),b_(2),b_(3) are in H.P. and let Delta=|(a_(1)-b_(1),a_(1)-b_(2),a_(1)-b_(3)),(a_(2)-b_(1),a_(2)-b_(2),a_(2)-b_(3)),(a_(3)-b_(1),a_(3)-b_(2),a_(3)-b_(3))| then prove that

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))

det[[1+a_(1),1,11,1+a_(2),11,1,1+a_(3)]]=a_(1)a_(2)a_(3)(1+(1)/(a_(1))+(1)/(a_(2))+(1)/(a_(3)))

If for an A.P.a_(1),a_(2),a_(3),......,a_(n),.........a_(1)+a_(3)+a_(5)=-12 and a_(1)a_(2)a_(3)=8 then the value of a_(2)+a_(4)+a_(6) equals