91Ó°ÊÓ

Write the first five terms of the sequence \(\left\\{a_{n}\right\\}\), \(n=0,1,2,3, \ldots\), and find \(\lim _{n \rightarrow \infty} a_{n}\). $$ a_{n}=\frac{1}{\sqrt{n+1}} $$

In Problems , find the population sizes for \(t=0,1,2, \ldots, 5\) for each recursion. $$ N_{t+1}=\frac{1}{2} N_{t} \text { with } N_{0}=4096 $$

In Problems , find the population sizes for \(t=0,1,2, \ldots, 5\) for each recursion. $$ N_{t+1}=\frac{1}{3} N_{t} \text { with } N_{0}=729 $$

Investigate the behavior of the discrete logistic equation $$ x_{t+1}=r x_{t}\left(1-x_{t}\right) $$ Compute \(x_{t}\) for \(t=0,1,2, \ldots, 20\) for the given values of \(r\) and \(x_{0}\), and graph \(x_{t}\) as a function of \(t\). \(r=3.8, x_{0}=0.5\)

Write the first five terms of the sequence \(\left\\{a_{n}\right\\}\), \(n=0,1,2,3, \ldots\), and find \(\lim _{n \rightarrow \infty} a_{n}\). $$ a_{n}=\frac{(-1)^{n}}{n+1} $$

Write the first five terms of the sequence \(\left\\{a_{n}\right\\}\), \(n=0,1,2,3, \ldots\), and find \(\lim _{n \rightarrow \infty} a_{n}\). $$ a_{n}=\frac{(-1)^{n}}{n^{3}+3} $$

Investigate the behavior of the discrete logistic equation $$ x_{t+1}=r x_{t}\left(1-x_{t}\right) $$ Compute \(x_{t}\) for \(t=0,1,2, \ldots, 20\) for the given values of \(r\) and \(x_{0}\), and graph \(x_{t}\) as a function of \(t\). \(r=3.8, x_{0}=0.1\)

In Problems , find the population sizes for \(t=0,1,2, \ldots, 5\) for each recursion. $$ N_{t+1}=\frac{1}{3} N_{t} \text { with } N_{0}=3645 $$

In Problems \(45-52\), write the first five terms of the sequence \(\left\\{a_{n}\right\\}\), \(n=0,1,2,3, \ldots\), and determine whether \(\lim _{n \rightarrow \infty} a_{n}\) exists. If the limit exists, find it. $$ a_{n}=\frac{n^{2}}{n+1} $$

In Problems , find the population sizes for \(t=0,1,2, \ldots, 5\) for each recursion. $$ N_{t+1}=\frac{1}{5} N_{t} \text { with } N_{0}=31250 $$