Try this beautiful Subjective Sequence Problem appeared in ISI Entrance - 2015.
(b) For any integer \(k>0\), give an example of a sequence of \(k\) positive integers whose reciprocals are in arithmetic progression.
Sequence
Arithmetic Progression
IIT Mathemathematics by Asit Dasgupta
ISI UG Entrance - 2015 , Subjective problem number - 8
Try to prove using the following hints.
Assume , \(d\) is the common difference for the given AP.
Therefore,
\( d = \frac{1}{m_2} - \frac{1}{m_1} \leq \frac{1}{m_1+1} - \frac{1}{m_1} = d' (say)\)
[Equality holds when , \(m_2 = m_1 + 1 \) ]
Hence proceed.
Now , \( \frac{1}{m_k} = \frac{1}{m_1} + d(k-1) \leq \frac{1}{m_1} + d'(k-1)\)
Notice that till now we haven't used \(m_1 < m_2 < \ldots < m_k\) are positive integers.
So , \(\frac{1}{m_k} > 0 \).
Therefore , \(\frac{1}{m_1} + d'(k-1) > 0 \).
Now use \( d' = \frac{1}{m_1+1} - \frac{1}{m_1} \).
\[\frac{1}{m_1} + d'(k-1) > 0 \]
\[\Rightarrow \frac{1}{m_1} + ( \frac{1}{m_1+1} - \frac{1}{m_1})(k-1) > 0 \]
Proceed with the above inequality and get \(m_1 + 2 > k .\)
As \(m_1 < m_2 < \ldots < m_k\) ,
so \(\frac{1}{m_1}, \frac{1}{m_2}, \ldots, \frac{1}{m_k}\)
is a decreasing \(AP.\)
Think about the LCM of \(m_1 < m_2 < \ldots < m_k\) .
Then you proceed.
Suppose , \(L = LCM(m_1 < m_2 < \ldots < m_k).\)
Now multiply \(L\) with all the reciprocals
i.e. with \(\frac{1}{m_1}, \frac{1}{m_2}, \ldots, \frac{1}{m_k}.\)
Then observe the pattern and try get such a sequence.
E.g. if \(k=5\)
Take \(5,4,3,2,1\) and \(LCM(5,4,3,2,1)=60\).
So the AP : \(\frac{5}{60},\frac{4}{60} , \frac{3}{60} , \frac{2}{60}, \frac{1}{60}\)
with common difference \(\frac{1}{60}.\)
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