Progressions

Algebra Level 4

\[ \dfrac{48}{2\times3} + \dfrac{47}{3\times4} + \dfrac{46}{4\times5} + \cdots + \dfrac{1}{49\times50} \]

If the value of the expression above is also equal to

\[ \dfrac{51}2 + k \left( 1 + \dfrac12 + \dfrac13 + \cdots + \dfrac1{50} \right) \]

for some constant \(k\), find \(k\).

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