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A3

Part of 1985 Putnam

Problems(1)

Putnam 1985 A3

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8/5/2019
Let dd be a real number. For each integer m≥0,m \geq 0, define a sequence {am(j)},j=0,1,2,…\left\{a_{m}(j)\right\}, j=0,1,2, \ldots by the condition \begin{align*} a_{m}(0)&=d / 2^{m},\\ a_{m}(j+1)&=\left(a_{m}(j)\right)^{2}+2 a_{m}(j),   j \geq 0. \end{align*} Evaluate lim⁡n→∞an(n).\lim _{n \rightarrow \infty} a_{n}(n).
Putnam