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Harvard-MIT Mathematics Tournament

Algebra Difficulty 4.5 AIME Find the answer United States

Problem:

Let $F_{n}$ be the Fibonacci sequence, that is, $F_{0}=0$ , $F_{1}=1$ , and $F_{n+2}=F_{n+1}+F_{n}$ . Compute $\sum_{n=0}^{\infty} F_{n} / 10^{n}$ .

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

Solution:

Write $F(x)=\sum_{n=0}^{\infty} F_{n} x^{n}$ . Then the Fibonacci recursion tells us that $F(x)-x F(x)-x^{2} F(x)=x$ , so $F(x)=x /\left(1-x-x^{2}\right)$ . Plugging in $x=1 / 10$ gives the answer.

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