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\begin{document}
\section*{Homework 1 - College Savings Plan \\ 
 Due: February 10} 
\textbf{Instructions:} Please answer the following questions with well thought out answers. Write your answers on a separate sheet of paper. STAPLE this sheet to the front of your answers. It is important to be able to explain ideas clearly. In this assignment pretend that you are trying to explain to a friend your answers. You should strive for your answers to be: precise, accurate, succinct, and understandable.
%
%\begin{question} 
%\end{question} 
In this activity you will mathematically describe a college savings plan. We begin by developing some Excel skills. 
\vspace{.2 in} 

\textbf{Recursive Relationships in Excel} 
\begin{defn} A \textbf{cell} is one entry in the Excel spreadsheet. 
\end{defn} 
\begin{enumerate} 
\item Find the Excel document on my website entitled CollegeSavingsPlan.xlsx and save it as CollegeSavingsPlan.yourlastname.yourfirstname.xlsx
\item The table Recursive Relationships 1 shows a geometric sequence. 
\begin{itemize} 
\item What is the relationship between $a_n$ and $a_{n+1}$? 
\item What is the first term? 
\item Write an explicit formula for $a_n$. 
\item In the table we see that the second term is $15$. Click on the $15$ and look at how that cell is ``computed''. Describe what you see. 

\end{itemize} 
\item The table Recursive Relationships 2 shows the beginning of an arithmetic sequence. 
\begin{itemize} 
\item What is the relationship between $a_n$ and $a_{n+1}$? 
\item What is the first term? 
\item Write an explicit formula for $a_n$. 
\item In the table we see that the second term is $14$. Click on the cell that has the 14 in it. You will see that this cell is computed by \textbf{= L9+2}. This means is that this cell contains the value of the cell $L9$ plus $2$. 
\item After you click on the cell that has the 14, the border of the cell will become highlighted and there will be a small square in the lower right hand corner. Click and drag the small square. As you drag the square you will be selecting are region of cells. Drag the square and select the next five cells. Each cell in that region is now two more than the one above it. 
\end{itemize} 
\item The table Recursive Relationships 3 shows the first term of a sequence. In the sequence, the following relationship holds: $a_{n+1}=4\cdot a_n+2$.
\begin{itemize} 
\item Is the sequence arithmetic or geometric? 
\item Make use of what we learned in Part 3 to complete the table (write a formula and drag it to the subsequent cells). 
\end{itemize} 
\end{enumerate} 

\newpage
\textbf{Basic Interest}
\begin{enumerate} 

\item Record your answers from the parts below in the table labeled \textbf{Basic Savings Plan}. In this basic plan you will deposit a sum only once. 

\item If you invest $\$500$ for one year at $7 \%$ interest, what is your balance after one year? 
\item If you invest $\$500$ for two years at $7 \%$ interest, what is your balance after two years? 
\item If your invest $\$500$ for three years at $7 \%$ interest, what is your balance after three years? 
\item If you invest $\$500$ for 18 years at $7 \%$ interest, what is your balance after 18 years? 
\item Assume you invest $\$500$ at $7 \%$ interest. Let $b_n$ be the balance after $n$ years. 
\begin{itemize} 
\item What is the relationship between $b_n$ and $b_{n+1}$? 
\item Is the sequence arithmetic or geometric? 
\item Write a formula for $b_n$ in terms of $n$. 
\item How does your formula for $b_n$ capture what you witnessed in Parts \ref{compare1}, \ref{compare2}, and \ref{compare3}? 
\item Write a formula in cell \textbf{G13} that calculates the ending balance. After you write the formula in \textbf{G13}, highlight the cell again and drag the black box down to \textbf{G20}. 
\item Use this to answer the questions below. 
\end{itemize} 
\item\label{compare1} Compare your balance after 18 years if you invested $\$500$ versus $\$1000$ at $7 \%$ interest. Your initial investment doubled. Did your ending balance double? 
\item\label{compare2} Compare your balance after 18 years if you invested $\$500$ at $7 \%$ interest versus  at $14 \%$ interest. Your rate of return doubled. Did your ending balance double? 
\item\label{compare3} Compare your balance if you invested  $\$500$ at $7 \%$ for $18$ years versus the same investment for $9$ years. The amount of time you invested has halved. Was your ending balance also reduced by half? 

\end{enumerate} 

\textbf{Reoccurring Deposits}

 Instead of just putting $\$500$ into an account and letting it grow, you may want to make reoccurring deposits.  
\begin{enumerate} 

\item If you initially invest $\$500$ and at the end of every year you deposit $\$100$ at $7 \%$ interest, what is your balance after one year? (check your answer with mine).
\item If you initially invest $\$500$ and at the end of every year you deposit $\$100$ at $7 \%$ interest, what is your balance after two years? 
\item If you initially invest $\$500$ and at the end of every year you deposit $\$100$ at $7 \%$ interest, what is your balance after four years? (hint: you should calculate the balance after 3 years first)
\item If you initially invest $\$500$ and at the end of every year you deposit $\$100$, then the total amount of money deposited forms a sequence. 
\begin{itemize} 
\item What type of sequence is this? 
\item Write down the recursive relationship.
\item Write down an explicit formula for the total amount of money deposited after $n$ years. 
\end{itemize} 
\item Assume you initially invest $\$500$ and at the end of every year you deposit $\$100$ at $7 \%$ interest. Let $d_n$ be the balance after $n$ years. 
\begin{itemize} 
\item What is the relationship between $d_n$ and $d_{n+1}$? 
\item Use an Excel formula to calculate the balance after five years knowing the balance after three years. 
\item Drag this formula down to calculate the balance after $18$ years. 
\end{itemize} 
\item Use the Excel technique developed in the previous question to compute the balance, if you initially invest $\$500$ and at the end of every year you deposit $\$200$ at $7 \%$ interest, what is your balance after four years? 
\end{enumerate} 

\textbf{Tuition}
The current tuition and fee schedule for a freshmen at the University of Utah taking 12 credits is shown in the following chart: 
\begin{center} 
\begin{tabular} { l  l  } 

Utah Resident & $\$3,140.87 $ \\ 

& \\ 

Non-Utah Resident & $  \$9,942.30 $\\ 

\end{tabular} 
\begin{enumerate} 
\item How many 12 credit semesters will you need to take to finish your major? 
\item How much money will this cost (assume that tuition is constant)? 
\item Use the Excel spreadsheet \textbf{Basic Savings Plan} you created to find how much money you would need to invest at $7\%$ for it to be worth what you calculated tuition will cost after 18 years. A guess and check method works here; try to get within $\$20$. 

\item Use the Excel spreadsheet \textbf{Reoccurring Deposits} to find an initial investment and a reoccurring deposit that will provide the balance you need after $20$ years. A guess and check method works here; try to get within $\$20$.  
\end{enumerate} 
\end{center} 
\end{document} 
