Convergent sequences of real numbers

Define the sequence of rational numbers p(n) recursively as follows:

\[p(1)=1\]

\[p(n)=1-\frac{p(n-1)}{2}\]

  • Compute and plot p(n) for \(n=1,2,\ldots , 20\). What inference do you draw about the terms p(n)?

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Define the sequence of rational numbers a(n) recursively as follows:

\[a(1)=1\]

\[a(n)=\frac{1+1}{(1+a(n-1))}\]

  • Compute a(n) and \(a(n)^2-2\) for \(n=1,2,\ldots , 20\). What inference do you draw about the terms \(a(n)^2-2\)?

    The graph is different with points that are increasing and decreasing. \(a(n)=\frac{1+1}{(1+a(n-1))} \) is a horizontal line while \(a(n)^2-2\) is a more complex solution with a different inf(a(n)) and sup(a(n)).