Chapter 2

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

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

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 \(t\). \(r=3.8, x_{0}=0.5\)

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

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

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. $$ N_{t+1}=\frac{1}{3} N_{t} \text { with } N_{0}=3645 $$

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 \(t\). \(r=3.8, x_{0}=0.9\)

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

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. $$ a_{n}=\frac{n^{2}}{n+1} $$

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 \(t\). \(r=3.8, x_{0}=0\)

In Problems , find the population sizes for \(t=0,1,2, \ldots, 5\) for each recursion. $$ N_{t+1}=\frac{1}{4} N_{t} \text { with } N_{0}=8192 $$

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. $$ a_{n}=\frac{n^{3}}{n+1} $$

$$ \text { In Problems } \text { , write } N_{t} \text { as a function of } t \text { for each recursion } $$ $$ N_{t+1}=2 N_{t} \text { with } N_{0}=15 $$

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. $$ a_{n}=\sqrt{n} $$

Graph the Ricker's curve $$ N_{t+1}=N_{t} \exp \left[R\left(1-\frac{N_{t}}{K}\right)\right] $$ in the \(N_{t}-N_{t+1}\) plane for the given values of \(R\) and \(\bar{K} .\) Find the points of intersection of this graph with the line \(N_{t+1}=N_{t}\). R=3, K=15

$$ \text { In Problems } \text { , write } N_{t} \text { as a function of } t \text { for each recursion } $$ $$ N_{t+1}=2 N_{t} \text { with } N_{0}=7 $$

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. $$ a_{n}=n^{2} $$

Graph the Ricker's curve $$ N_{t+1}=N_{t} \exp \left[R\left(1-\frac{N_{t}}{K}\right)\right] $$ in the \(N_{t}-N_{t+1}\) plane for the given values of \(R\) and \(\bar{K} .\) Find the points of intersection of this graph with the line \(N_{t+1}=N_{t}\). R=2.5, K=12

$$ \text { In Problems } \text { , write } N_{t} \text { as a function of } t \text { for each recursion } $$ $$ N_{t+1}=3 N_{t} \text { with } N_{0}=12 $$

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. $$ a_{n}=2^{n} $$

Graph the Ricker's curve $$ N_{t+1}=N_{t} \exp \left[R\left(1-\frac{N_{t}}{K}\right)\right] $$ in the \(N_{t}-N_{t+1}\) plane for the given values of \(R\) and \(\bar{K} .\) Find the points of intersection of this graph with the line \(N_{t+1}=N_{t}\). R=4, K=20

$$ \text { In Problems } \text { , write } N_{t} \text { as a function of } t \text { for each recursion } $$ $$ N_{t+1}=3 N_{t} \text { with } N_{0}=3 $$

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. $$ a_{n}=\left(\frac{1}{2}\right)^{n} $$

Investigate the behavior ofthe Ricker's curve $$ N_{t+1}=N_{f} \exp \left[R\left(1-\frac{N_{t}}{K}\right)\right] $$ Compute \(N_{t}\) for \(t=1,2, \ldots, 20\) for the given values of \(R, K\), and \(N_{0}\), and graph \(N_{t}\) as a function of \(t\). (a) \(R=1, K=20, N_{0}=5\) (b) \(R=1, K=20, N_{0}=10\) (c) \(R=1, K=20, N_{0}=20\) (d) \(R=1, K=20, N_{0}=0\)

$$ \text { In Problems } \text { , write } N_{t} \text { as a function of } t \text { for each recursion } $$ $$ N_{t+1}=4 N_{t} \text { with } N_{0}=24 $$

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. $$ a_{n}=3^{n} $$

Investigate the behavior ofthe Ricker's curve $$ N_{t+1}=N_{f} \exp \left[R\left(1-\frac{N_{t}}{K}\right)\right] $$ Compute \(N_{t}\) for \(t=1,2, \ldots, 20\) for the given values of \(R, K\), and \(N_{0}\), and graph \(N_{t}\) as a function of \(t\). (a) \(R=1.8, K=20, N_{0}=5\) (b) \(R=1.8, K=20, N_{0}=10\) (c) \(R=1.8, K=20, N_{0}=20\) (d) \(R=1.8, K=20, N_{0}=0\)

$$ \text { In Problems } \text { , write } N_{t} \text { as a function of } t \text { for each recursion } $$ $$ N_{t+1}=5 N_{t} \text { with } N_{0}=17 $$

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. $$ a_{n}=\left(\frac{1}{3}\right)^{n} $$

$$ \text { In Problems } \text { , write } N_{t} \text { as a function of } t \text { for each recursion } $$ $$ N_{t+1}=\frac{1}{2} N_{t} \text { with } N_{0}=5000 $$

Formal Definition of Limits: In Problems \(53-64, \lim _{n \rightarrow \infty} a_{n}=a .\) Find the limit \(a\), and determine \(N\) so that \(\left|a_{n}-a\right|<\epsilon\) for all \(n>N\) for the given value of \(\epsilon\). $$ a_{n}=\frac{1}{n}, \epsilon=0.01 $$

Investigate the behavior ofthe Ricker's curve $$ N_{t+1}=N_{f} \exp \left[R\left(1-\frac{N_{t}}{K}\right)\right] $$ Compute \(N_{t}\) for \(t=1,2, \ldots, 20\) for the given values of \(R, K\), and \(N_{0}\), and graph \(N_{t}\) as a function of \(t\). (a) \(R=2.8, K=20, N_{0}=5\) (b) \(R=2.8, K=20, N_{0}=10\) (c) \(R=2.8, K=20, N_{0}=20\) (d) \(R=2.8, K=20, N_{0}=0\)

$$ \text { In Problems } \text { , write } N_{t} \text { as a function of } t \text { for each recursion } $$ $$ N_{t+1}=\frac{1}{2} N_{t} \text { with } N_{0}=2300 $$

Formal Definition of Limits: \lim _{n \rightarrow \infty} a_{n}=a .\( Find the limit \)a\(, and determine \)N\( so that \)\left|a_{n}-a\right|<\epsilon\( for all \)n>N\( for the given value of \)\epsilon$. $$ a_{n}=\frac{1}{n}, \epsilon=0.02 $$

Compute \(N_{t}\) and \(N_{t} / N_{t-1}\) for \(t=2,3,4, \ldots, 20\) when $$ N_{t+1}=N_{t}+N_{t-1} $$ with \(N_{0}=1\) and \(N_{1}=1\).

$$ \text { In Problems } \text { , write } N_{t} \text { as a function of } t \text { for each recursion } $$ $$ N_{t+1}=\frac{1}{3} N_{t} \text { with } N_{0}=8000 $$

Formal Definition of Limits: \lim _{n \rightarrow \infty} a_{n}=a .\( Find the limit \)a\(, and determine \)N\( so that \)\left|a_{n}-a\right|<\epsilon\( for all \)n>N\( for the given value of \)\epsilon$. $$ a_{n}=\frac{1}{n^{2}}, \epsilon=0.01 $$

Compute \(N_{t}\) and \(N_{t} / N_{t-1}\) for \(t=2,3,4, \ldots .20\) when $$ N_{t+1}=N_{t}+2 N_{t-1} $$ with \(N_{0}=1\) and \(N_{1}=1\).

$$ \text { In Problems } \text { , write } N_{t} \text { as a function of } t \text { for each recursion } $$ $$ N_{t+1}=\frac{1}{3} N_{t} \text { with } N_{0}=3500 $$

Formal Definition of Limits: \lim _{n \rightarrow \infty} a_{n}=a .\( Find the limit \)a\(, and determine \)N\( so that \)\left|a_{n}-a\right|<\epsilon\( for all \)n>N\( for the given value of \)\epsilon$. $$ a_{n}=\frac{1}{n^{2}}, \epsilon=0.001 $$

Formal Definition of Limits: \lim _{n \rightarrow \infty} a_{n}=a .\( Find the limit \)a\(, and determine \)N\( so that \)\left|a_{n}-a\right|<\epsilon\( for all \)n>N\( for the given value of \)\epsilon$. $$ a_{n}=\frac{1}{\sqrt{n}}, \epsilon=0.1 $$

$$ \text { In Problems } \text { , write } N_{t} \text { as a function of } t \text { for each recursion } $$ $$ N_{t+1}=\frac{1}{7} N_{t} \text { with } N_{0}=6400 $$

Formal Definition of Limits: \lim _{n \rightarrow \infty} a_{n}=a .\( Find the limit \)a\(, and determine \)N\( so that \)\left|a_{n}-a\right|<\epsilon\( for all \)n>N\( for the given value of \)\epsilon$. $$ a_{n}=\frac{1}{\sqrt{n}}, \epsilon=0.05 $$

Formal Definition of Limits: \lim _{n \rightarrow \infty} a_{n}=a .\( Find the limit \)a\(, and determine \)N\( so that \)\left|a_{n}-a\right|<\epsilon\( for all \)n>N\( for the given value of \)\epsilon$. $$ a_{n}=\frac{(-1)^{n}}{n}, \epsilon=0.01 $$

In Problems , graph the line \(N_{t+1}=R N_{t}\) in the \(N_{t}-N_{t+1}\) plane for the indicated value of \(R\) and locate the points \(\left(N_{t}, N_{t+1}\right), t=0\), 1\. and 2. for the given value of \(N_{0}\). $$ R=2, N_{0}=3 $$

Formal Definition of Limits: \lim _{n \rightarrow \infty} a_{n}=a .\( Find the limit \)a\(, and determine \)N\( so that \)\left|a_{n}-a\right|<\epsilon\( for all \)n>N\( for the given value of \)\epsilon$. $$ a_{n}=\frac{(-1)^{n}}{n}, \epsilon=.001 $$

Formal Definition of Limits: \lim _{n \rightarrow \infty} a_{n}=a .\( Find the limit \)a\(, and determine \)N\( so that \)\left|a_{n}-a\right|<\epsilon\( for all \)n>N\( for the given value of \)\epsilon$. $$ a_{n}=\frac{n}{n+1}, \epsilon=0.01 $$

In Problems , graph the line \(N_{t+1}=R N_{t}\) in the \(N_{t}-N_{t+1}\) plane for the indicated value of \(R\) and locate the points \(\left(N_{t}, N_{t+1}\right), t=0\), 1\. and 2. for the given value of \(N_{0}\). $$ R=4, N_{0}=2 $$