- #1

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## Homework Statement

Let ##x_k=k## for ##k \leq 31## and ##\displaystyle x_{k+1}=\frac{x_1+x_2+........x_k}{k}## for ##k \geq 31##. Also let ##y_k=x_k## for ##k \leq 31## and ##\displaystyle y_{k+1}=\frac{y_k+y_{k-1}+........y_{k-30}}{31}## for ##k \geq 31##. Now if ##z_k=y_k-x_k## for all ##k ε N##. Find ##\lim_{n→∞} z_n##.

## Homework Equations

## The Attempt at a Solution

I figured out that ##x_{k+1}=x_{k+2}=....=16##, so the question reduces to

[tex]\displaystyle \lim_{n→∞} z_n=y_n-16[/tex]

I am having trouble finding ##\lim_{n→∞}=y_n##.

I plugged in some numbers in the expression of ##y_k## starting k=31.

When k=31, ##y_{32}=16##.

When k=32, ##y_{33}=\frac{15+16*31}{31}##

The next terms go even more big. I am stuck here.

Any help is appreciated. Thanks!