- #1

Saitama

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## Homework Statement

A sequence of numbers ##x_n## is determined by the equality ##x_n=\frac{x_{n-1}+x_{n-2}}{2}## and the values of ##x_0## and ##x_1##. Compute ##x_n## in terms of ##x_0, x_1## and ##n##. Also prove that $$\lim_{n \rightarrow \infty} x_n=\frac{x_0+2x_1}{3}$$.

## Homework Equations

## The Attempt at a Solution

I don't know where to start with such kind of problem so I tried with calculating a few terms.

$$x_2=\frac{x_0+x_1}{2}$$

$$x_3=\frac{x_0+3x_1}{4}$$

$$x_4=\frac{3x_0+5x_1}{8}$$

$$x_5=\frac{5x_0+11x_1}{16}$$

Okay, I see the denominator is ##2^{n-1}## but I can't catch any pattern for the numerator. :(

Any help is appreciated. Thanks!