If `a_(1),a_(2),a_(3)anda_(4)` be the coefficients of four consecutive terms in the expansion of `(1+x)^(n)`, then prove that `(a_(1))/(a_(1)+a_(2)),(a_(2))/(a_(2)+a_(3))and(a_(3))/(a_(3)+a_(4))` are in A.P.

If a_(r) be the coefficient of x^(r) in the expression of (1+bx^(2)+cx^(3))^(n),"then prove that" , a_(3)=nc

If the coefficients of the consicutive four terms in the expansion of (1+x)^(n)" be "a_(1),a_(2),a_(3)and a_(4) respectively , show that , (a_(1))/(a_(1)+a_(2))+(a_(3))/(a_(3)+a_(4))=2.(a_(2))/(a_(2)+a_(3)).

Find the coordinates of the centriod of the triangle whose vertices are ( a_(1), b_(1), c_(1)) , (a_(2), b_(2), c_(2)) and (a_(3), b_(3), c_(3)) .

If a_(r)= (cos 2r pi + i sin 2r pi)^(1/9) . Then the value of |(a_(1),a_(2),a_(3)),(a_(4),a_(5),a_(6)),(a_(7),a_(8),a_(9))| is

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_(n) are in H.P., prove that, a_(1)a_(2) + a_(2)a_(3) + a_(3)a_(4) +…+ a_(n-1)a_(n) = (n-1)a_(1)a_(n)

A_(1), A_(2) and A_(3) are three events. Show that the simultaneous occurrence of the events is P(A_(1) cap A_(2) cap A_(3))=P(A_(1))P(A_(2)//A_(1))P[A_(3)//(A_(1) cap A_(2))] State under which condition P(A_(1) cap A_(2) cap A_(3))=P(A_(1))P(A_(2))P(A_(3))

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),a_(2),a_(3),a_(4),...,a_(n) are n positive real numbers whose product is a fixed number c, then the minimum value of a_(1)+ a_(2) + ...+a_(n-1)+2a_(n) is-

If A_(1), A_(2),..,A_(n) are any n events, then