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putnam120 · #5 September 17th, 2007, 09:46 PM

well ill start by proving my first assertion $\lim_{n\to\infty}\frac{a_n}{a_{n-1}}=\phi$

Let $L=\lim_{n\to\infty}\frac{a_n}{a_{n-1}}$ . Then after expanding $a_n$ by the recursion we have

$L=\lim_{n\to\infty}\frac{a_{n-1}+a_{n-2}}{a_{n-1}}=\lim_{n\to\infty}1+\frac{a_{n-2}}{a_{n-1}}$ . Notice that the second term is just the reciprocal of the original limit so we solve

$L=1+\frac1L\Longrightarrow L^2-L-1=0\Longrightarrow L=\frac{1\pm\sqrt{5}}{2}$ . We choose the positive solution because all the terms in the sequence are obviously positive and thus a negative ratio is not possible.