Let x1,x2,..., xn be positive real numbers. Prove that
n^2 <= (x1 + x2 + ... + xn ) ( 1/x1 + 1/x2 + ... + 1/xn )
and
(x1 + x2 + ... + xn)/(sqrt(n)) <= sqrt( x1^2 + x2^2 +...+ xn^2)
Still not able to solve the second one, any help appreciated.
I was able to solve the first one using induction and the lemma from another thread i asked about that (x/y) + (y/x) > 2
Using induction, you can factor the sequence of k + 1 in to the sequence up to k, and (x/y) + (y/x) for each where x = x1, x2,...,xk and y = x(k+1). Since there are k of those factors and each is greater than 2, we have the first two factors are great than k^2 + 2k. Also, when factoring you get a factor of 2, so with K^2 + 2k + 2 > K^2 + 2k + 1 = (K+1)^2, the theorem is proved by induction.