It Telescopes?

Calculus Level 3

Let \(\{a_{n}\}\) be a sequence of real numbers satisfying \[ \begin{cases} a_{0}=1 \\ a_{n+1}=\sqrt{4+3a_{n}+a_{n}^{2}}-2 & \text{for } n \ge 0. \end{cases} \] Let \(\displaystyle S=\sum_{n=0}^{\infty} a_{n}\).

  • If \(S\) converges, submit your answer as \(\big\lfloor 100S \big\rfloor\).
  • If \(S\) diverges, submit your answer as \(-1\).

This problem is based on a recent Putnam contest problem.

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