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Magazine Problem 98
The sequence \(\mid a_{n}\) \\} is recursively defined. Compute \(a_{n}\) for \(n=1,2, \ldots, 5\) $$ a_{n+1}=\sqrt{a_{n}}, a_{0}=16 $$
Problem 99
The sequence \(\mid a_{n}\) \\} is recursively defined. Compute \(a_{n}\) for \(n=1,2, \ldots, 5\) $$ a_{n+1}=a_{n}+\frac{1}{a_{n}}, a_{0}=1 $$
Problem 100
The sequence \(\mid a_{n}\) \\} is recursively defined. Compute \(a_{n}\) for \(n=1,2, \ldots, 5\) $$ a_{n+1}=2 a_{n}^{2}, a_{0}=1 $$
Problem 101
In Problems 101-110, the sequence \(\left\\{a_{n} \mid\right.\) is recursively defined. Find all fixed points of \(\left[a_{n}\right\\} .\) $$ a_{n+1}=\frac{1}{2} a_{n}+2 $$
Problem 102
The sequence \(\left\\{a_{n} \mid\right.\) is recursively defined. Find all fixed points of \(\left[a_{n}\right\\} .\) $$ a_{n+1}=\frac{1}{3} a_{n}+\frac{4}{3} $$
Problem 103
The sequence \(\left\\{a_{n} \mid\right.\) is recursively defined. Find all fixed points of \(\left[a_{n}\right\\} .\) $$ a_{n+1}=\frac{5}{2}-\frac{1}{2} a_{n} $$
Problem 104
The sequence \(\left\\{a_{n} \mid\right.\) is recursively defined. Find all fixed points of \(\left[a_{n}\right\\} .\) $$ a_{n+1}=a_{n}^{2}-a_{n} $$
Problem 109
The sequence \(\left\\{a_{n} \mid\right.\) is recursively defined. Find all fixed points of \(\left[a_{n}\right\\} .\) $$ a_{n+1}=\sqrt{5 a_{n}} $$
Problem 110
The sequence \(\left\\{a_{n} \mid\right.\) is recursively defined. Find all fixed points of \(\left[a_{n}\right\\} .\) $$ a_{n+1}=\sqrt{a_{n}+2} $$
Problem 111
In Problems 111-118, assume that \(\lim _{n \rightarrow \infty} a_{n}\) exists. Find all fixed points of \(\left\\{a_{n}\right\\}\), and use a table or other reasoning to guess which fixed point is the limiting value for the given initial condition. $$ a_{n+1}=\frac{1}{3} a_{n}+\frac{4}{3}, a_{0}=0 $$
