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Three Sequences

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September 7, 2024
algebrafloor function2024

Problem Statement

Let {an}n=0∞\{ a_n \}_{n=0}^{\infty}, {bn}n=0∞\{ b_n \}_{n=0}^{\infty}, and {cn}n=0∞\{ c_n \}_{n=0}^{\infty} be sequences of real numbers such that for all k≥1k\geq 1, \begin{align*} a_k&=\left\lfloor \sqrt{2}+\frac{k-1}{2024} \right\rfloor+a_{k-1} \\ b_k+c_k&=1 \\ a_{k-1}b_k&=a_kc_k. \end{align*} Suppose that a0=1a_0=1, b0=2b_0=2, and c0=3c_0=3. Given that 2≈1.4142\sqrt2\approx1.4142, compute ∑k=12024(akbk−ak−1ck). \sum_{k=1}^{2024}(a_kb_k-a_{k-1}c_k).
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