Let be k positive numbers such that their reciprocals are in A.P. Show that
. Also find such a sequence for positive integer k > 0.
……
Teacher: This is a simple application of Arithmetic Mean – Harmonic Mean Inequality. First notice that if we have nothing to prove as
,
is definitely greater than k.
So we are interested in the cases where
Now apply the A.M. – H.M. inequality
Student: Ok.
Now we already know that are in A.P. So sum of these k terms in A.P. is (first term plus last term) times number of terms by 2. In other words
So our expression becomes
This implies
I cannot think what to do next.
Teacher: It is quite simple actually. Notice that are integers then their reciprocals are less than (or at most equal to) 1.
Hence .
and
are different hence both cannot be 1.
Thus
Hence since
Now k is at least 3 (we are already done with k < 3). So