This is Problem number 7 from the ISI Subjective Entrance Exam based on the Arithmetic Sequence of reciprocals. Try to solve the problem.
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
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
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.
Now k is at least 3 (we are already done with k < 3). So