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# Arithmetic Sequence of reciprocals (ISI subjective 2015 Problem 7)

Let $$m_1, m_2 , … , m_k$$ be k positive numbers such that their reciprocals are in A.P. Show that $$k< m_1 + 2$$ . 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 $$k \le 2$$ we have nothing to prove as $$m_1 \ge 1$$ , $$m_1 +2$$ is definitely greater than k.

So we are interested in the cases where $$k \ge 3$$
Now apply the A.M. – H.M. inequality

Student: Ok. $$\displaystyle { \frac {k}{\frac{1}{m_1} + \frac {1}{m_2} + … + \frac{1}{m_k}} \ge \frac{m_1 + m_2 + … + m_k}{k} }$$

Now we already know that $$\displaystyle { \frac{1}{m_1} , \frac {1}{m_2} , … , \frac{1}{m_k} }$$ 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 $$\displaystyle { \frac{1}{m_1} + \frac {1}{m_2} + … + \frac{1}{m_k} = \frac {k}{2} \left ( \frac{1}{m_1} + \frac{1}{m_k} \right)}$$

So our expression becomes $$\displaystyle { \frac {k}{ \frac {k}{2} \left ( \frac{1}{m_1} + \frac{1}{m_k} \right)} \le \frac{m_1 + m_2 + … + m_k}{k} }$$

This implies $$\displaystyle { k \le (m_1 + m_2 + … + m_k) \frac {\left ( \frac{1}{m_1} + \frac{1}{m_k} \right)}{2} }$$

I cannot think what to do next.

Teacher: It is quite simple actually. Notice that $$m_1 , … , m_k$$ are integers then their reciprocals are less than (or at most equal to) 1.

Hence $$\displaystyle { \frac {1}{m_1} \le 1 , \frac{1}{m_k} \le 1 \Rightarrow \frac{1}{m_1} + \frac{1} {m_k} \le 2 }$$. $$m_1$$ and $$m_k$$ are different hence both cannot be 1.

Thus $$\displaystyle { \frac{1}{m_1} + \frac{1} {m_k} < 2 \Rightarrow \frac {\left ( \frac{1}{m_1} + \frac{1}{m_k} \right)}{2} < 1 }$$

Hence $$\displaystyle { k \le (m_1 + m_2 + … + m_k) \frac {\left ( \frac{1}{m_1} + \frac{1}{m_k}\right)}{2} < (m_1 + m_2 + … + m_k) \le m_1 + k-1}$$ since $$1 \le m_2 , … , 1 \le m_k$$

Now k is at least 3 (we are already done with k < 3). So $$k < m_1 + k-1 \le m_1 + 2$$

June 8, 2015