Skip to main content
Contents Index Dark Mode Prev Up Next \(\newcommand{\im}{{\mathrm{im}}}
\newcommand{\given}{{\, :\,}}
\newcommand{\hanoi}{\mathrm{hanoi}}
\newcommand{\lcm}{{\mathrm{lcm}}}
\newcommand{\ID}{{\mathrm{ID}}}
\newcommand{\chr}{{\mathtt{chr}}}
\newcommand{\ord}{{\mathtt{ord}}}
\newcommand{\set}[1]{\left\lbrace #1 \right\rbrace}
\newcommand{\seq}[2]{( #1 )_{#2}}
\newcommand{\gr}[1]{\left( #1 \right)}
\newcommand{\abs}[1]{\left| #1 \right|}
\newcommand{\size}[1]{\# #1}
\newcommand{\id}[1]{\mathrm{id}_{#1}}
\newcommand{\ip}[1]{\left\langle #1 \right\rangle}
\newcommand{\floor}[1]{\left\lfloor #1\right\rfloor}
\newcommand{\ceiling}[1]{\left\lceil #1\right\rceil}
\newcommand{\indemph}[1]{{\textbf{\emph{#1}}\index{#1}}}
\renewcommand{\implies}{\to}
\renewcommand{\iff}{\leftrightarrow}
\newcommand{\lxor}{\oplus}
\newcommand{\cP}{\wp}
\newcommand{\pset}{\cP}
\newcommand{\ZZ}{\mathbb{Z}}
\newcommand{\QQ}{\mathbb{Q}}
\newcommand{\NN}{\mathbb{N}}
\newcommand{\CC}{\mathbb{C}}
\newcommand{\RR}{\mathbb{R}}
\newcommand{\barr}[1]{\overline{#1}}
\newcommand{\sage}{\textsc{Sage}}
\newcommand{\magma}{\textsc{Magma}}
\newcommand{\bob}{\text{Bob}}
\newcommand{\true}{\text{T}}
\newcommand{\false}{\text{F}}
\newcommand{\ahat}{\hat{a}}
\newcommand{\ah}{\ahat}
\newcommand{\bhat}{\hat{b}}
\newcommand{\abar}{\bar{a}}
\newcommand{\cA}{\mathcal{A}}
\newcommand{\cF}{\mathcal{F}}
\newcommand{\cW}{\mathcal{W}}
\newcommand{\todo}[1]{{\color{red} TODO: #1}}
\renewcommand{\div}{\operatorname{div}}
\newcommand{\gexp}[3]{#1^{#2 #3}}
\newcommand{\gexpp}[3]{\displaystyle\left(#1\right)^{#2 #3}}
\newcommand{\glog}[3]{\log_{#1}^{#3}#2}
\def\dbend{{!!!}}
\newcommand{\lt}{<}
\newcommand{\gt}{>}
\newcommand{\amp}{&}
\definecolor{fillinmathshade}{gray}{0.9}
\newcommand{\fillinmath}[1]{\mathchoice{\colorbox{fillinmathshade}{$\displaystyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\textstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptscriptstyle\phantom{\,#1\,}$}}}
\)
Exercises 2.4.6 Exercises
1.
Give the definition for these terms. Be sure to set up any notation that is required.
geometric progression, common ratio, initial term
arithmetic progression, common difference, initial term
solution of a recurrence relation
2.
Compute the initial term and common ratio of the following geometric progression.
\begin{equation*}
16, 8, 4, 2, 1, \frac{1}{2}, \frac{1}{4}, \frac{1}{8},
\dots
\end{equation*}
3. Compute
\({\sum_{i = 1}^{5} i}\text{.}\) Simplify.
4. Compute
\({\sum_{i = 1}^{100}(3i + 7)}\text{.}\) Simplify.
5. Compute
\({\sum_{k = 50}^{100} k^2}\text{.}\) Simplify.
6. Compute
\({\sum_{k = 0}^{10} 3 \cdot 2^k}\text{.}\) Simplify.
7.
What are the terms \(a_1\text{,}\) \(a_2\text{,}\) \(a_3\text{,}\) and \(a_4\) of each of these sequences \(\seq{a_n}{n}\text{?}\)
\(\displaystyle a_n = 1\)
\(\displaystyle a_n = \floor{\frac{n}{2}}\)
\(\displaystyle a_n = (-1)^n\)
\(\displaystyle a_n = 2(-3)^{n + 1} + 4\)
\(\displaystyle a_n = (-\frac{1}{2})^n\)
8.
What are the terms \(a_1\text{,}\) \(a_2\text{,}\) \(a_3\text{,}\) and \(a_4\) of each of these sequences \(\seq{a_n}{n}\text{?}\)
\(\displaystyle a_n = 1 + (-1)^n\)
\(\displaystyle a_n = -(-3)^n\)
\(\displaystyle a_n = 3n + 4\)
\(\displaystyle a_n = 3^n - 2^n\)
\(\displaystyle a_n = \sqrt{\floor{n}}\)
9.
Show that each of these sequences \(\seq{a_n}{n}\) is a solution of the recurrence relation
\begin{equation*}
a_n = -3a_{n - 1} + 4a_{n - 2}.
\end{equation*}
\(\displaystyle a_n = 0\)
\(\displaystyle a_n = 1\)
\(\displaystyle a_n = (-4)^n\)
\(\displaystyle a_n = 2(-4)^n + 3\)
10.
Suppose you are hired at company in this year with a starting salary of \(\${50000}\text{.}\) Every year, you receive a \(5\%\) raise plus an additional \(\${1000}\text{.}\)
Set up a recurrence relation for your salary after
\(n\) years.
What is your salary in 10 years?
Find an explicit formula for your salary in
\(n\) years.
