Leet Sum

Algebra Level 5

The sequence \(\{a_k\}_{k=1}^{112}\) satisfies \(a_1 = 1\) and \(a_n = \dfrac{1337 + n}{a_{n-1}}\), for all positive integers \(n\). Let

\[S = \left\lfloor a_{10}a_{13} + a_{11}a_{14} + a_{12}a_{15} + \cdots + a_{109}a_{112}\right\rfloor. \]

Find the remainder when \(S\) is divided by \(1000\).

This problem is posed by Akshaj.

Details and assumptions

The function \(\lfloor x \rfloor: \mathbb{R} \rightarrow \mathbb{Z}\) refers to the greatest integer smaller than or equal to \(x\). For example \(\lfloor 2.3 \rfloor = 2\) and \(\lfloor -5 \rfloor = -5\).

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