91Ó°ÊÓ

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. $$ a_{n+1}=\sqrt{2 a_{n}}, a_{0}=1 $$

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. $$ a_{n+1}=\frac{3}{a_{n}+2}, a_{0}=0 $$

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. $$ a_{n+1}=2 a_{n}\left(1-a_{n}\right), a_{0}=0.1 $$

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. $$ a_{n+1}=\frac{1}{2}\left(a_{n}+\frac{9}{a_{n}}\right), a_{0}=-1 $$

In Problems 119-124, write each sum in expanded form. $$ \sum_{k=1}^{4} \sqrt{k} $$

Write each sum in expanded form. $$ \sum_{k=3}^{5}(k-1)^{2} $$

Write each sum in expanded form. $$ \sum_{k=2}^{6} 3^{k} $$

Write each sum in expanded form. $$ \sum_{k=1}^{3} \frac{k^{2}}{k^{2}+1} $$

Write each sum in expanded form. $$ \sum_{n=0}^{3} a_{n} \text { where } a_{0}=1 \text { and } a_{n+1}=2 a_{n} $$

Write each sum in expanded form. $$ \sum_{n=0}^{3} a_{n} \text { where } a_{0}=1 \text { and } a_{n+1}=2 a_{n} $$