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Learn MoreThis 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.**

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*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

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