If `a ,a_1, a_2, a_3, a_(2n),b` are in A.P. and `a ,g_1,g_2,g_3, ,g_(2n),b` . are in G.P. and `h` s the H.M. of `aa n db ,` then prove that `(a_1+a_(2n))/(g_1g_(2n))+(a_2+a_(2n-1))/(g_1g_(2n-1))++(a_n+a_(n+1))/(g_ng_(n+1))=(2n)/h`

if a,a_(1),a_(2),a_(3),......,a_(2n),b are in A.P. and a,g_(1),g_(2),............g_(2n),b are in G.P. and h is H.M. of a,b then (a_(1)+a_(2n))/(g_(1)*g_(2n))+(a_(2)+a_(2n-1))/(g_(2)*g_(2n-1))+.........+(a_(n)+a_(n+1))/(g_(n)*g_(n+1)) is equal

If a,a_(1),a_(2),...,a_(10),b are in A.P.and a,g_(1),g_(2),...,g_(10),b are in G.P.and h is the H.M.betweein a and b,then (a_(1)+a_(2)+...+a_(10))/(g_(1)g_(10))+(a_(2)+a_(3)+...+a_(9))/(g_(2)g_(9))+......+(a_(5)+a_(6))/(g_(5)g_(6))

The sequence, a, a1,a 2. b is an A.P; the sequence a,g1,g2 .... g2n,b is a GP. h is the H.M of a, b. The series (a_1 +a_2n)/(g_1g_2n) + .......+(a_n+a_(n+1))/(g_n*g_(n+1))

In any G.P. , prove that a_(n-p) xxa_(n+p) = (a_(n))^(2) .

If 1, a_1,a_2,a_3 ,…, a_(n-1) are the nth roots of unity then prove that : (1-a_1)(1-a_2)(1-a_3)...(1-a_(n-1)) =n.

If a,A_(1),A_(2),b are in A.P., a,G_(1),G_(2),b are in G.P. and a, H_(1),H_(2),b are in H.P., then the value of G_(1)G_(2)(H_(1)+H_(2))-H_(1)H_(2)(A_(1)+A_(2)) is

If 1, a_1,a_2,a_3 ,…, a_(n-1) are the nth roots of unity then prove that : 1+a_1+a_2+…+a_(n-1) =0.

If a_(1),a_(2),a_(3)….a_(n) are positive and (n-1)s = a_(1)+a_(2)+….+ a_(n) then prove that a_(1),a_(2),a_(3)…a_(n) ge (n-1)^(n)(s-a_(1))(s-a_(2))….(s-a_(n))

If a_1,a_2,a_3,.....,a_n are in AP, prove that 1/(a_1a_2)+1/(a_2a_3)+1/(a_3a_4)+...+1/(a_(n-1)a_n)=(n-1)/(a_1a_n) .