# Solution: 2013-16 Limit of a sequence

For real numbers $$a, b$$, find the following limit.
$\lim_{n \to \infty} n \left( 1 – \frac{a}{n} – \frac{b \log (n+1)}{n} \right)^n.$

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A real sequence $$x_1, x_2, x_3, \cdots$$ satisfies the relation $$x_{n+2} = x_{n+1} + x_n$$ for $$n = 1, 2, 3, \cdots$$. If a number $$r$$ satisfies $$x_i = x_j = r$$ for some $$i$$ and $$j$$ $$(i \neq j)$$, we say that $$r$$ is a repeated number in this sequence. Prove that there can be more than $$2013$$ repeated numbers in such a sequence, but it is impossible to have infinitely many repeated numbers.