C L's sequence

Algebra Level 5

Define:

\[ \begin{align}a_n &= 1 + \frac 1 2 + \ldots + \frac 1 n\\ b_n &= a_1 + a_2 + \ldots + a_n\\ c_n &= \frac {b_1} 2 + \frac {b_2} 3 + \ldots + \frac {b_n}{n+1}\end{align} \]

If \(j\) and \(k\) are integers between -1000 and 1000 such that \( c_{123} = j \cdot a_{124} + k\), what is the value of \(j- k\)?

This problem is shared by C L, adapted from USAMO.

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