C L's sequence

Algebra Level 5

Define:

an=1+12++1nbn=a1+a2++ancn=b12+b23++bnn+1 \begin{aligned}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{aligned}

If jj and kk are integers between -1000 and 1000 such that c123=ja124+k c_{123} = j \cdot a_{124} + k, what is the value of jkj- k?

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

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