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) , cdots a_(2n), b are in A.P. and a, g_(1) , g_(2) , g_(3) , cdots , g_(2n), b are in G.P. and h is the H.M of a and b then prove that (a_(1)+a_(2n))/(g_(1)g_(2n))+(a_(2)+a_(2n-1))/(g_(2)g_(2n-1))+cdots+ (a_(n)+a_(n+1))/(g_(n)g_(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_(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_1, a_2, ,a_n >0, then prove that (a_1)/(a_2)+(a_2)/(a_3)+(a_3)/(a_4)++(a_(n-1))/(a_n)+(a_n)/(a_1)> n

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))

If a_1, a_2,...... ,a_n >0, then prove that (a_1)/(a_2)+(a_2)/(a_3)+(a_3)/(a_4)+.....+(a_(n-1))/(a_n)+(a_n)/(a_1)> n

If a_1, a_2,...... ,a_n >0, then prove that (a_1)/(a_2)+(a_2)/(a_3)+(a_3)/(a_4)+.....+(a_(n-1))/(a_n)+(a_n)/(a_1)> n

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_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