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Magazine Problem 96
The sequence \(\left\\{a_{n}\right\\}\) is recursively defined. Find all fixed points of \(\left\\{a_{n}\right\\}\) $$ a_{n+1}=-\frac{1}{3} a_{n}+\frac{1}{4} $$
Problem 97
The sequence \(\left\\{a_{n}\right\\}\) is recursively defined. Find all fixed points of \(\left\\{a_{n}\right\\}\) $$ a_{n+1}=\frac{4}{a_{n}} $$
Problem 98
The sequence \(\left\\{a_{n}\right\\}\) is recursively defined. Find all fixed points of \(\left\\{a_{n}\right\\}\) $$ a_{n+1}=\frac{7}{a_{n}} $$
Problem 99
The sequence \(\left\\{a_{n}\right\\}\) is recursively defined. Find all fixed points of \(\left\\{a_{n}\right\\}\) $$ a_{n+1}=\frac{2}{a_{n}+2} $$
Problem 101
The sequence \(\left\\{a_{n}\right\\}\) is recursively defined. Find all fixed points of \(\left\\{a_{n}\right\\}\) $$ a_{n+1}=\sqrt{5 a_{n}} $$
Problem 102
The sequence \(\left\\{a_{n}\right\\}\) is recursively defined. Find all fixed points of \(\left\\{a_{n}\right\\}\) $$ a_{n+1}=\sqrt{7 a_{n}} $$
Problem 103
In Problems 103-110, assume that \(\lim _{n \rightarrow \infty} a_{n}\) exists. Find all fixed ooints of \(\left\\{a_{n}\right\\}\), and use a table or other reasoning to guess which ixed point is the limiting value for the given initial condition. $$ a_{n+1}=\frac{1}{2}\left(a_{n}+5\right), a_{0}=1 $$
Problem 107
Assume that \(\lim _{n \rightarrow \infty} a_{n}\) exists. Find all fixed ooints of \(\left\\{a_{n}\right\\}\), and use a table or other reasoning to guess which ixed point is the limiting value for the given initial condition. $$ a_{n+1}=2 a_{n}\left(1-a_{n}\right), a_{0}=0.1 $$
Problem 109
Assume that \(\lim _{n \rightarrow \infty} a_{n}\) exists. Find all fixed ooints of \(\left\\{a_{n}\right\\}\), and use a table or other reasoning to guess which ixed point is the limiting value for the given initial condition. $$ a_{n+1}=\frac{1}{2}\left(a_{n}+\frac{4}{a_{n}}\right), a_{0}=1 $$
