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\begin{document}
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\centerline{\bf Math 365 -- Wednesday 2/6/19}
\centerline{\bf  Section 2.4: Sequences and summations}

\begin{try}~
\begin{enumerate}[(a)]
\item For each of the following sequences, compute the terms $a_0$, $a_1$, $a_2$, and $a_3$.
\begin{enumerate}[(a)]
\item $a_n = 3$;
\item $a_n = 7+4^n$;
\item $a_n = 2^n+(-2)^n$.
\end{enumerate}



\item For each of the following sequences defined by recurrence relations and initial conditions, answer the following.
\begin{enumerate}[(a)]
\item  Compute  the first four terms ($a_0$, $a_1$, $a_2$, and $a_3$).
\item Decide if $\{a_n\}$ is arithmetic, geometric, or neither. If it is arithmetic or geometric, then find a closed formula for $a_n$. 
\end{enumerate}
\begin{enumerate}[(i)]
\item The sequence satisfying $a_0 = 2$ and $a_n = \frac{1}{2} a_{n-1}$. 
\item The sequence satisfying $a_0 = -1$ and $a_n = a_{n-1}+5$. 
\item The sequence satisfying $a_0 = 1$, $a_1 = -1$ and $a_n = a_{n-2}* a_{n-1}$. 
\item The sequence satisfying $a_0 = 2$ and $a_n = -a_{n-1}$. 
\end{enumerate}




\item Find at least three different sequences beginning with the terms $3, 5, 7$ whose terms are generated by a simple formula or rule.

\item For the following sequences, try to find the pattern. Decide if they are arithmetic, geometric, or neither. If it's arithmetic or geometric, find a closed formula expressing the $n$th term of the sequence. 
\begin{enumerate}[(i)]
\item $5, 1, -3, -7, -11, \dots$
\item $1, 4, 9, 16, 25, \dots$
\item $3, 9, 27, 81, 243, \dots$
\end{enumerate}

\item Show that both of the following sequences are solutions to the recurrence
relation $a_n = -3a_{n-1} + 4 a_{n-2}$ with initial condition $a_0 = 1$.

\begin{enumerate}[(i)]
\item $a_n=1$;
\item $a_n=(-4)^n$.

\end{enumerate}
\end{enumerate}
\end{try}



\medskip


\begin{try} 

\begin{enumerate}[(a)]
\item Compute the first three partial sums for the following infinite series.
\begin{enumerate}[(i)]
\item $\displaystyle \sum_{i=0}^\infty 5i + 1$;
\item $\displaystyle \sum_{i=4}^\infty i(i+1)$;
\end{enumerate}

\item Calculate the following.
\begin{enumerate}[(i)]
	\item $\displaystyle \sum_{j=0}^8 (1 + (-1)^j)$

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	\item $\displaystyle \sum_{j=-1}^2 \sum_{i = 2}^{3} (2i + 3j)$

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	\item $\displaystyle \sum_{j \in \{-1,4,15\}} 2 $

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	\item $\displaystyle \sum_{j \in \left\{z \in \ZZ ~\big|~ |z| \leq 2\right\}} j$

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	\item $\displaystyle \sum_{n=2}^5 a_n - a_{n-1}$ where $a_n= n!$ 

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	\item $\displaystyle \sum_{i=1}^{2000} i$ 
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	\item $\displaystyle \sum_{i=0}^{10} \frac{2}{3^i}$ 
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	\item $\displaystyle \sum_{i=0}^{\infty} \frac{2}{3^i}$ 
\end{enumerate}
\end{enumerate}
\end{try}


\begin{try}A little more. 
\begin{enumerate}[(a)]
\item For each of these sequences find a recurrence relation satisfied by this sequence. (The answers are not unique because there are infinitely many different recurrence relations satisfied by any sequence.)

\begin{enumerate}[(i)]
\item $a_n=2n+3$;
\item $a_n=5^n$;
\item $a_n=n^2$.
\end{enumerate}

 
\item Use partial sums to explain why, for any sequence $a_0, a_1, \dots, a_n$, that 
$$\sum_{i = 1}^n a_i - a_{i-1} = a_n - a_0.$$
[Let $S_n = \sum_{i = 1}^n a_i - a_{i-1}$. Write out $S_1, S_2, S_3,$ and so on until you see the pattern. Then use the fact that $S_n = S_{n-1} + (a_n - a_{n-1})$.]

\item Read in section 2.4 about product notation $\prod_{i=m}^n a_i$. Then, what are the values of the following products?
\begin{enumerate}[(a)]
\item $\displaystyle \prod_{i=0}^{10} i $

\medskip

\item $\displaystyle \prod_{i=5}^8 i(i+1) $

\medskip

\item $\displaystyle \prod_{i=1}^{100} (-1)^i$


\end{enumerate}

\item Express $n!$ using product notation.





\end{enumerate}



\end{try}

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