If a1, a2,...... ,an >0, then prove tha...

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`

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If a_1,a_2,a3,...,a_n are in A.P then show that 1/(a_1a_2)+1/(a_2a_3)+1/(a_3a_4)+...+1/(a_(n-1)a_n)=(n-1)/(a_1a_n)

If the sequence a_1, a_2, a_3,....... a_n ,dot forms an A.P., then prove that a_1^2-a_2^2+a_3^2-a_4^2+.......+ a_(2n-1)^2 - a_(2n)^2=n/(2n-1)(a_1^2-a_(2n)^2)

If a_1,a_2,.....a_n are in H.P., then the expression a_1a_2 + a_2a_3 + ... + a_(n-1)a_n is equal to

If a_1. a_2 ....... a_n are positive and (n - 1) s = a_1 + a_2 +.....+a_n then prove that (a_1 + a_2 +....+a_n)^n ge (n^2 - n)^n (s - a_1) (s - a_2)........(s - a_n)

If S=a_1+a_2+......+a_n,a_i in R^+ for i=1 to n, then prove that S/(S-a_1)+S/(S-a_2)+......+S/(S-a_n) ge n^2/(n-1), AA n ge 2

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_1, a_2,a_3, ,a_n is an A.P. with common difference d , then prove that "tan"[tan^(-1)(d/(1+a_1a_2))+tan^(-1)(d/(1+a_2a_3))+......+tan^(-1)(d/(1+a_(n-1)a_n))]=((n-1)d)/(1+a_1a_n)

Let a_1,a_2 ,a_3 ...... be an A.P. Prove that sum_(n=1)^(2m)(-1)^(n-1)a_n^2=m/(2m-1)(a_1^2-a_(2m)^2) .

If the coefficients of four consecutive terms in the expansion of (1+x)^n are a_1,a_2,a_3 and a_4 respectively. then prove that a_1/(a_1+a_2)+a_3/(a_3+a_4)=2a_2/(a_2+a_3).

If a_1,a_2, a_3, a_4 be the coefficient of four consecutive terms in the expansion of (1+x)^n , then prove that: (a_1)/(a_1+a_2)+(a_3)/(a_3+a_4)=(2a_2)/(a_2+a_3)dot

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