Exercises

1. Graph the given terms of each of the following sequences. 
   1. \(13,~14,~15,~16,~17, \ldots\)
   2. \(3,~6,~9,~12,~15, \ldots\)
   3. \(5,~9,~13,~17,~21, \ldots\)
2. Consider the following graph of the first few terms in a sequence:
   
   1. Use the graph to indicate the term value for term number \(8\).
   2. Use the graph to indicate the term that has a value of \(13\).
   3. Would this graph provide an efficient way to find the term that has the value of \(100\)?  Explain.
   4. What is the general term of the sequence? Use the general term to find the term that has the value of \(100\).
3. Nicholas opened a savings account on June 1st and made an initial deposit right away and then added a particular amount to the account on the last day of each month.  The following table shows the account balance each month after his deposit has been made.  

   Month	Balance (\(\bf $\))
   \(1\)	\(650\)
   \(2\)	\(800\)
   \(3\)	\(950\)
   \(4\)	\(1100\)

   1. What was Nicholas' initial deposit, and how much money did he add to the account each month after that?
   2. Represent this information in a graph.
   3. Nicholas is saving for a vacation that will cost \($2000\). After how many months will he have enough money in the account for the vacation? Use your graph to find this value.
4. Tyriana is building a tower as shown.  
   
   1. Graph the sequence of horizontal blocks.
   2. On the same graph as from part a), graph the sequence of vertical blocks.
   3. Explain how you can obtain the graph of the sequence of total blocks.  
5. Consider the following three sequences:
   Sequence 1
   \(8,~9,~10,~11,\ldots\)

   Sequence 2
   \(2,~6,~10,~14,\ldots\)

   Sequence 3
   \(7,~11,~15,~19,\ldots \)
   1. Represent Sequence 1 and Sequence 2 as graphs on the same axes.  Is there a term where both sequences have the same value?  Explain.
   2. Represent Sequence 1 and Sequence 3 as graphs on the same axes.  Is there a term where both sequences have the same value?  Explain.
6. In a terrarium, there are \(10\) crickets and \(1\) lizard at the end of the first day. The next morning, \(3\) more crickets are placed in the terrarium and the lizard eats \(4\) crickets throughout the day.  This pattern continues each day.  
   1. Create a table to show how many crickets are in the terrarium at the end of each day, over \(6\) days.
   2. Create a graph to display this information.
   3. How is this graph different from the graphs we have seen in this lesson?
   4. Draw a line through the data points on your graph, and extend it to the right.  Does this line accurately represent what happens in the terrarium on the \(7^{th}\) day and afterwards?
7. A wading pool is being filled with a hose at a rate of \(50\) mL/second. Unfortunately, the pool also has a hole in it. While it is being filled, water is also leaking out of the pool with \(5\) mL leaking out in the first second, \(6\) mL in the next second, \(7\) mL in the third second, and so on.
   1. For the first few seconds, more water is entering the pool than is leaving the pool.  Does this ever change?  If so, when?
   2. If the pool has a capacity of \(50\) L, then will it ever fill up?  If so, after how many seconds?
8. Consider the following three conditions describing two different sequences. Let \(n\) represent the term number.
   • Sequence 1 has a general term that looks like  \(n+a\), where \(a\) is some integer.
     (For example, \(n+4\) is of this form where \(a\) is the integer \(4\).)
   • Sequence 2 has a general term of the form \(b \times n\), where \(b\) is some integer.
     (For example, \(7\times n\) is of this form where \(b\) is the integer \(7\).)
   • Sequence 1 and Sequence 2 have the same value at term number \(5\).
   Create two sequences that satisfy the above conditions.  How can a graph help you to solve this problem?