<FlowDocument PagePadding="5,0,5,0" AllowDrop="False"><Paragraph Margin="0,0,0,0" TextAlignment="Left" FontFamily="Times New Roman" FontSize="14.6666666666667"><Run FontFamily="Arial" FontSize="18"></Run></Paragraph><Paragraph Margin="0,0,0,0" TextAlignment="Left" FontFamily="Times New Roman" FontSize="14.6666666666667"><Run FontFamily="Arial"></Run><Image Stretch="None"><Image.Source><BitmapImage UriSource="211f32c1cd4d4281a99eaedc66bf9569.png" CacheOption="OnLoad" /></Image.Source></Image><Run FontFamily="Arial" FontSize="18"></Run></Paragraph><Paragraph Margin="0,0,0,0" TextAlignment="Left" FontFamily="Times New Roman" FontWeight="Bold" FontSize="14.6666666666667"><Run FontFamily="Arial" FontSize="18">Sigma Notation and Geometric Sequences</Run></Paragraph><Paragraph Margin="0,0,0,0" TextAlignment="Left" FontFamily="Times New Roman" FontSize="14.6666666666667"><Run FontFamily="Arial" FontSize="18"></Run></Paragraph><Paragraph Margin="0,0,0,0" TextAlignment="Left" FontFamily="Times New Roman" FontSize="14.6666666666667"><Run FontFamily="Arial" FontSize="18">In this lesson, we will continue our study of series and take a look at a way to write them using formal mathematical notation. We will also explore another special type of sequence.</Run></Paragraph><Paragraph Margin="0,0,0,0" TextAlignment="Left" FontFamily="Times New Roman" FontSize="14.6666666666667"><Run FontFamily="Arial" FontSize="18"></Run></Paragraph><Paragraph Margin="0,0,0,0" TextAlignment="Left" FontFamily="Times New Roman" FontWeight="Bold" FontSize="14.6666666666667"><Run FontFamily="Arial" FontSize="18">Sigma Notation</Run></Paragraph><Paragraph Margin="0,0,0,0" TextAlignment="Left" FontFamily="Times New Roman" FontWeight="Bold" FontSize="14.6666666666667"><Run FontFamily="Arial" FontSize="18"></Run></Paragraph><Paragraph Margin="0,0,0,0" TextAlignment="Left" FontFamily="Times New Roman" FontSize="14.6666666666667"><Run FontFamily="Arial" FontSize="18">Since series can get quite tedious to write out each time we want to use them, and they are so frequently used, there is special notation that is used to represent them. Consider the sequence of terms </Run><Image Stretch="None"><Image.Source><BitmapImage UriSource="3c9aa34c2a9140bf9f4c0e6b53a5d14d.png" CacheOption="OnLoad" /></Image.Source></Image><Run FontFamily="Arial" FontSize="18">. We can write this sequence using what is known as </Run><Run FontFamily="Arial" FontStyle="Italic" FontWeight="Bold" FontSize="18">summation notation</Run><Run FontFamily="Arial" FontSize="18">, or </Run><Run FontFamily="Arial" FontStyle="Italic" FontWeight="Bold" FontSize="18">sigma notation</Run><Run FontFamily="Arial" FontSize="18">. The symbol used is the Greek letter </Run><Image Stretch="None"><Image.Source><BitmapImage UriSource="693fc6551c53497a98636cda230b9c27.png" CacheOption="OnLoad" /></Image.Source></Image><Run FontFamily="Arial" FontSize="18">, and the sum of the terms in the sequence we’re looking at is written as follows:</Run></Paragraph><Paragraph Margin="0,0,0,0" TextAlignment="Left" FontFamily="Times New Roman" FontSize="14.6666666666667"><Run FontFamily="Arial" FontSize="18"></Run></Paragraph><BlockUIContainer TextAlignment="Justify"><Image Stretch="None"><Image.Source><BitmapImage UriSource="d4c2b8cb7cf74cc79428104c91a31203.png" CacheOption="OnLoad" /></Image.Source></Image></BlockUIContainer><Paragraph Margin="0,0,0,0" TextAlignment="Center" FontFamily="Times New Roman" FontSize="14.6666666666667"><Run FontFamily="Arial" FontSize="18"></Run></Paragraph><Paragraph Margin="0,0,0,0" TextAlignment="Left" FontFamily="Times New Roman" FontSize="14.6666666666667"><Run FontFamily="Arial" FontSize="18">Notice that we did not use </Run><Run FontFamily="Arial" FontStyle="Italic" FontSize="18">n</Run><Run FontFamily="Arial" FontSize="18"> as the subscript of the general term (which is known as the </Run><Run FontFamily="Arial" FontWeight="Bold" FontSize="18">index of the series</Run><Run FontFamily="Arial" FontSize="18">). This is because we want to reserve </Run><Run FontFamily="Arial" FontStyle="Italic" FontSize="18">n</Run><Run FontFamily="Arial" FontSize="18"> to be part of the subscript of the</Run><Run FontFamily="Arial" FontStyle="Italic" FontSize="18"> final term </Run><Run FontFamily="Arial" FontSize="18">in our sum. So, we’ve chosen the letter </Run><Run FontFamily="Arial" FontStyle="Italic" FontSize="18">k</Run><Run FontFamily="Arial" FontSize="18"> instead. Other common choices for the index are </Run><Run FontFamily="Arial" FontStyle="Italic" FontSize="18">i</Run><Run FontFamily="Arial" FontSize="18"> and </Run><Run FontFamily="Arial" FontStyle="Italic" FontSize="18">j</Run><Run FontFamily="Arial" FontSize="18">, but you can choose any letter you want, as long as it doesn’t conflict with other letters you’ve defined.</Run></Paragraph><Paragraph Margin="0,0,0,0" TextAlignment="Left" FontFamily="Times New Roman" FontSize="14.6666666666667"><Run FontFamily="Arial" FontSize="18"></Run></Paragraph><Paragraph Margin="0,0,0,0" TextAlignment="Left" FontFamily="Times New Roman" FontSize="14.6666666666667"><Run FontFamily="Arial" FontSize="18">The notation contains four important parts:</Run></Paragraph><Paragraph Margin="0,0,0,0" TextAlignment="Left" FontFamily="Times New Roman" FontSize="14.6666666666667"><Run FontFamily="Arial" FontSize="18"></Run></Paragraph><List MarkerStyle="Decimal" Margin="0,0,0,0" Padding="0,0,0,0"><ListItem Margin="48,0,0,0"><Paragraph Margin="0,0,0,0" TextAlignment="Left" FontFamily="Times New Roman" FontSize="14.6666666666667"><Run FontFamily="Arial" FontSize="18">The </Run><Run FontFamily="Arial" FontStyle="Italic" FontSize="18">sigma symbol</Run><Run FontFamily="Arial" FontSize="18"> represents that this is a sum of terms.</Run></Paragraph></ListItem><ListItem Margin="48,0,0,0"><Paragraph Margin="0,0,0,0" TextAlignment="Left" FontFamily="Times New Roman" FontSize="14.6666666666667"><Run FontFamily="Arial" FontSize="18">The symbols at the bottom of the sigma symbol tells us which value we want to assign to the index first.</Run></Paragraph></ListItem><ListItem Margin="48,0,0,0"><Paragraph Margin="0,0,0,0" TextAlignment="Left" FontFamily="Times New Roman" FontSize="14.6666666666667"><Run FontFamily="Arial" FontSize="18">The symbol at the top of the sigma symbol tells us the last value for the index.</Run></Paragraph></ListItem><ListItem Margin="48,0,0,0"><Paragraph Margin="0,0,0,0" TextAlignment="Left" FontFamily="Times New Roman" FontSize="14.6666666666667"><Run FontFamily="Arial" FontSize="18">The term </Run><Image Stretch="None"><Image.Source><BitmapImage UriSource="7ff921f169b24bfba05ab764458d8b64.png" CacheOption="OnLoad" /></Image.Source></Image><Run FontFamily="Arial" FontSize="18"> tells us what the general rule (form) is for each of the terms in the series.</Run></Paragraph></ListItem></List><Paragraph Margin="0,0,0,0" TextAlignment="Left" FontFamily="Times New Roman" FontSize="14.6666666666667"><Run FontFamily="Arial" FontSize="18"></Run></Paragraph><BlockUIContainer TextAlignment="Justify"><Image Stretch="None"><Image.Source><BitmapImage UriSource="1c60b2f8c7694e27a6fdb6af630a6c41.png" CacheOption="OnLoad" /></Image.Source></Image></BlockUIContainer><Paragraph Margin="0,0,0,0" TextAlignment="Left" FontFamily="Times New Roman" FontSize="14.6666666666667"><Run FontFamily="Arial" FontSize="18"></Run></Paragraph><Paragraph Margin="0,0,0,0" TextAlignment="Left" FontFamily="Times New Roman" FontSize="14.6666666666667"><Run FontFamily="Arial" FontSize="18"></Run><Image Stretch="None"><Image.Source><BitmapImage UriSource="38ce33a460d7488a96b99658e968d734.png" CacheOption="OnLoad" /></Image.Source></Image><Run FontFamily="Arial" FontWeight="Bold" FontSize="18">Note: </Run><Run FontFamily="Arial" FontSize="18">The smallest value for the index does not need to be 0. The series can begin with any number for the index. Therefore, there are multiple ways to write a series using this notation.</Run></Paragraph><Paragraph Margin="0,0,0,0" TextAlignment="Left" FontFamily="Times New Roman" FontSize="14.6666666666667"><Run FontFamily="Arial" FontSize="18"></Run></Paragraph><Paragraph Margin="0,0,0,0" TextAlignment="Left" FontFamily="Times New Roman" FontWeight="Bold" FontSize="14.6666666666667"><Run FontFamily="Arial" FontSize="18"></Run></Paragraph><Paragraph Margin="0,0,0,0" TextAlignment="Left" FontFamily="Times New Roman" FontSize="14.6666666666667"><Run FontFamily="Arial" FontWeight="Bold" FontSize="18">Example #1: </Run><Run FontFamily="Arial" FontSize="18">We write each of the following series in expanded form. </Run></Paragraph><Paragraph Margin="0,0,0,0" TextAlignment="Left" FontFamily="Times New Roman" FontSize="14.6666666666667"><Run FontFamily="Arial" FontSize="18"></Run></Paragraph><Paragraph Margin="0,0,0,0" TextAlignment="Left" FontFamily="Times New Roman" FontSize="14.6666666666667"><Run FontFamily="Arial" FontSize="18">a.</Run></Paragraph><Paragraph Margin="0,0,0,0" TextAlignment="Left" FontFamily="Times New Roman" FontSize="21.3333333333333"><Run FontFamily="Arial" FontSize="18"></Run><LineBreak /><Image Stretch="None"><Image.Source><BitmapImage UriSource="7dec1c3e915e40f386a25160b3d2ef22.png" CacheOption="OnLoad" /></Image.Source></Image></Paragraph><Paragraph Margin="0,0,0,0" TextAlignment="Left" FontFamily="Times New Roman" FontSize="14.6666666666667"><Run FontFamily="Arial" FontSize="18"></Run></Paragraph><Paragraph Margin="0,0,0,0" TextAlignment="Left" FontFamily="Times New Roman" FontSize="14.6666666666667"><Run FontFamily="Arial" FontSize="18">Notice that there are </Run><Run FontFamily="Arial" FontWeight="Bold" FontSize="18">7</Run><Run FontFamily="Arial" FontSize="18"> terms, even though 6 is the last value of the index. Since we started with 0, we have 7 total terms in the series.</Run><Run FontFamily="Arial" FontSize="18"></Run><LineBreak /><Run FontFamily="Arial" FontSize="18" Foreground="#FF00B050"></Run></Paragraph><Paragraph Margin="0,0,0,0" TextAlignment="Left" FontFamily="Times New Roman" FontSize="14.6666666666667"><Run FontFamily="Arial" FontSize="18">b.</Run><Run FontFamily="Arial" FontSize="18" Foreground="#FF00B050"></Run><Run FontFamily="Arial" FontSize="18"></Run><LineBreak /><Image Stretch="None"><Image.Source><BitmapImage UriSource="9f6f525bc6ca487b82d0127a2d24c4cd.png" CacheOption="OnLoad" /></Image.Source></Image></Paragraph><Paragraph Margin="0,0,0,0" TextAlignment="Left" FontFamily="Times New Roman" FontSize="14.6666666666667"><Run FontFamily="Arial" FontSize="18"></Run></Paragraph><Paragraph Margin="0,0,0,0" TextAlignment="Left" FontFamily="Times New Roman" FontSize="14.6666666666667"><Run FontFamily="Arial" FontSize="18">c.</Run><Run FontFamily="Arial" FontSize="18" Foreground="#FF00B050"></Run><Run FontFamily="Arial" FontSize="18"></Run><LineBreak /><Image Stretch="None"><Image.Source><BitmapImage UriSource="254e0906ae564c82b852ac4eee1c94bf.png" CacheOption="OnLoad" /></Image.Source></Image></Paragraph><Paragraph Margin="0,0,0,0" TextAlignment="Right" FontFamily="Times New Roman" FontSize="14.6666666666667"><Run FontFamily="Arial" FontSize="18"></Run></Paragraph><Paragraph Margin="0,0,0,0" TextAlignment="Left" FontFamily="Times New Roman" FontWeight="Bold" FontSize="14.6666666666667"><Run FontFamily="Arial" FontSize="18"></Run></Paragraph><Paragraph Margin="0,0,0,0" TextAlignment="Left" FontFamily="Times New Roman" FontSize="14.6666666666667"><Run FontFamily="Arial" FontWeight="Bold" FontSize="18">Example #2: </Run><Run FontFamily="Arial" FontSize="18">Write the following series in sigma notation. Try these on you own, and then check your answers below. Note, though, that your notation might differ from the ones in the solution depending on your first value for the index.</Run></Paragraph><Paragraph Margin="0,0,0,0" TextAlignment="Left" FontFamily="Times New Roman" FontSize="14.6666666666667"><Run FontFamily="Arial" FontSize="18"></Run></Paragraph><BlockUIContainer><Image Stretch="None"><Image.Source><BitmapImage UriSource="04e6cc8bc6cd4b9ea9a7fa87b288445a.png" CacheOption="OnLoad" /></Image.Source></Image></BlockUIContainer><Paragraph Margin="0,0,0,0" TextAlignment="Left" FontFamily="Times New Roman" FontSize="14.6666666666667"><Run FontFamily="Arial" FontSize="18"></Run></Paragraph><Paragraph Margin="0,0,0,0" TextAlignment="Left" FontFamily="Times New Roman" FontWeight="Bold" FontSize="14.6666666666667"><Run FontFamily="Arial" FontSize="18"></Run></Paragraph><Paragraph Margin="0,0,0,0" TextAlignment="Left" FontFamily="Times New Roman" FontWeight="Bold" FontSize="14.6666666666667"><Run FontFamily="Arial" FontSize="18">Solution: </Run></Paragraph><Paragraph Margin="0,0,0,0" TextAlignment="Left" FontFamily="Times New Roman" FontSize="18.6666666666667"><Run FontFamily="Arial" FontSize="18">a. </Run><Run FontFamily="Arial" FontSize="18"></Run><LineBreak /><Image Stretch="None"><Image.Source><BitmapImage UriSource="2cacd56f053d4d31863aee5100c455ed.png" CacheOption="OnLoad" /></Image.Source></Image></Paragraph><Paragraph Margin="0,0,0,0" TextAlignment="Left" FontFamily="Times New Roman" FontSize="14.6666666666667"><Run FontSize="18"></Run><LineBreak /><Run FontFamily="Arial" FontSize="18"></Run></Paragraph><Paragraph Margin="0,0,0,0" TextAlignment="Left" FontFamily="Times New Roman" FontSize="14.6666666666667"><Run FontFamily="Arial" FontSize="18">b. </Run></Paragraph><BlockUIContainer><Image Stretch="None"><Image.Source><BitmapImage UriSource="7801ab5bce4541b2ab1a2bfd758620db.png" CacheOption="OnLoad" /></Image.Source></Image></BlockUIContainer><Paragraph Margin="0,0,0,0" TextAlignment="Left" FontFamily="Times New Roman" FontSize="14.6666666666667"><Run FontFamily="Arial" FontSize="18">c. </Run></Paragraph><Paragraph Margin="0,0,0,0" TextAlignment="Left" FontFamily="Times New Roman" FontSize="14.6666666666667"><Run FontFamily="Arial"></Run><Image Stretch="None"><Image.Source><BitmapImage UriSource="f91b4ded0c4348e49a90ea78fa284a47.png" CacheOption="OnLoad" /></Image.Source></Image><Run FontSize="18"></Run><LineBreak /><Run FontFamily="Arial" FontSize="18"></Run></Paragraph><Paragraph Margin="0,0,0,0" TextAlignment="Left" FontFamily="Times New Roman" FontSize="14.6666666666667"><Run FontFamily="Arial" FontSize="18">d.</Run></Paragraph><BlockUIContainer><Image Stretch="None"><Image.Source><BitmapImage UriSource="7044e813866f4eb38bfb51efcdc58e38.png" CacheOption="OnLoad" /></Image.Source></Image></BlockUIContainer><Paragraph Margin="0,0,0,0" TextAlignment="Right" FontFamily="Times New Roman" FontWeight="Bold" FontSize="14.6666666666667"><Run FontFamily="Arial" FontSize="18"></Run></Paragraph><Paragraph Margin="0,0,0,0" TextAlignment="Right" FontFamily="Times New Roman" FontWeight="Bold" FontSize="14.6666666666667"><Run FontFamily="Arial" FontSize="18"></Run><Image Stretch="None"><Image.Source><BitmapImage UriSource="652e1f2ea7c44f3cb8d4df8cbced34d5.png" CacheOption="OnLoad" /></Image.Source></Image></Paragraph><Paragraph Margin="0,0,0,0" TextAlignment="Left" FontFamily="Times New Roman" FontSize="14.6666666666667"><Run FontFamily="Arial" FontSize="18"></Run></Paragraph><Paragraph Margin="0,0,0,0" TextAlignment="Left" FontFamily="Times New Roman" FontSize="14.6666666666667"><Run FontFamily="Arial"></Run><Image Stretch="None"><Image.Source><BitmapImage UriSource="4a45317337204406bd7a8b0ecd6ba740.png" CacheOption="OnLoad" /></Image.Source></Image><Run FontFamily="Arial" FontSize="18" Foreground="#FF00B050"></Run></Paragraph><Paragraph Margin="0,0,0,0" TextAlignment="Left" FontFamily="Times New Roman" FontWeight="Bold" FontSize="14.6666666666667"><Run FontFamily="Arial" FontSize="18">Vehicle Depreciation</Run></Paragraph><Paragraph Margin="0,0,0,0" TextAlignment="Left" FontFamily="Times New Roman" FontSize="14.6666666666667"><Run FontFamily="Arial" FontSize="18"></Run></Paragraph><Paragraph Margin="0,0,0,0" TextAlignment="Left" FontFamily="Times New Roman" FontSize="14.6666666666667"><Run FontFamily="Arial" FontSize="18">Have you ever heard the saying that new cars depreciate in value “as soon as you drive them off the lot?” This is actually true of any car; however, new cars depreciate at a greater average rate initially than used ones. Used cars depreciate at a steady rate of approximately 7-12% per year. Suppose you were looking at a used car for $13,000 that depreciates in value at a rate of 8% per year. How much will that car be worth in 9 years, when you expect you’ll be ready to buy your next car?</Run></Paragraph><Paragraph Margin="0,0,0,0" TextAlignment="Left" FontFamily="Times New Roman" FontSize="14.6666666666667"><Run FontFamily="Arial" FontSize="18"></Run></Paragraph><Paragraph Margin="0,0,0,0" TextAlignment="Left" FontFamily="Times New Roman" FontSize="14.6666666666667"><Run FontFamily="Arial" FontSize="18">Since the car is worth $13,000 when you buy it, at the beginning of the second year the car would be worth </Run><Image Stretch="None"><Image.Source><BitmapImage UriSource="71ba3f3deaee4473a9f38b7fea225804.png" CacheOption="OnLoad" /></Image.Source></Image><Run FontFamily="Arial" FontSize="18">. At the beginning of the third year, the car would be worth </Run><Image Stretch="None"><Image.Source><BitmapImage UriSource="73717c1148f7487bba885353ecf1e377.png" CacheOption="OnLoad" /></Image.Source></Image><Run FontFamily="Arial" FontSize="18">, and at the beginning of the fourth year, it would be worth </Run><Image Stretch="None"><Image.Source><BitmapImage UriSource="37b239585bce48faa95b097c2ee530fb.png" CacheOption="OnLoad" /></Image.Source></Image><Run FontFamily="Arial" FontSize="18">. So, if we consider this sequence of values that represent the worth of the car at the beginning of each year, we have </Run><Image Stretch="None"><Image.Source><BitmapImage UriSource="f6fff5a3825b41ec995606ca4bc0266f.png" CacheOption="OnLoad" /></Image.Source></Image><Run FontFamily="Arial" FontSize="18">. We can express this recursively, since each new term is defined using the previous term, and we do so as follows:</Run></Paragraph><Paragraph Margin="0,0,0,0" TextAlignment="Left" FontFamily="Times New Roman" FontSize="18.6666666666667"><Run FontFamily="Arial" FontSize="18" Foreground="#FF00B050"></Run><Run FontFamily="Arial" FontSize="18"></Run><LineBreak /><Image Stretch="None"><Image.Source><BitmapImage UriSource="8a187c99f46d42c3a94da2ee17647e15.png" CacheOption="OnLoad" /></Image.Source></Image></Paragraph><Paragraph Margin="0,0,0,0" TextAlignment="Left" FontFamily="Times New Roman" FontSize="14.6666666666667"><Run FontFamily="Arial" FontSize="18"></Run></Paragraph><Paragraph Margin="0,0,0,0" TextAlignment="Left" FontFamily="Times New Roman" FontSize="14.6666666666667"><Run FontFamily="Arial" FontSize="18"></Run></Paragraph><Paragraph Margin="0,0,0,0" TextAlignment="Left" FontFamily="Times New Roman" FontSize="14.6666666666667"><Run FontFamily="Arial" FontSize="18">After simplifying, we can write this as</Run></Paragraph><Paragraph Margin="0,0,0,0" TextAlignment="Left" FontFamily="Times New Roman" FontSize="14.6666666666667"><Run FontFamily="Arial" FontSize="18"></Run><Run FontFamily="Arial" FontSize="18"></Run><LineBreak /><Image Stretch="None"><Image.Source><BitmapImage UriSource="36090057ab044b97bb70661a1f43b481.png" CacheOption="OnLoad" /></Image.Source></Image></Paragraph><Paragraph Margin="0,0,0,0" TextAlignment="Left" FontFamily="Times New Roman" FontSize="14.6666666666667"><Run FontFamily="Arial" FontSize="18"></Run></Paragraph><Paragraph Margin="0,0,0,0" TextAlignment="Left" FontFamily="Times New Roman" FontSize="14.6666666666667"><Run FontFamily="Arial" FontSize="18">Our recursive formula above provides us with the relationship between each successive term as it relates to the previous one in our sequence. It tells us that each term is 0.92 of the previous term. We call this type of sequence a </Run><Run FontFamily="Arial" FontStyle="Italic" FontSize="18">geometric sequence</Run><Run FontFamily="Arial" FontSize="18">.</Run></Paragraph><Paragraph Margin="0,0,0,0" TextAlignment="Left" FontFamily="Times New Roman" FontSize="14.6666666666667"><Run FontFamily="Arial" FontSize="18"></Run></Paragraph><Table CellSpacing="0" Margin="0,0,0,0"><Table.Columns><TableColumn Width="750" /></Table.Columns><TableRowGroup><TableRow><TableCell Padding="7.2,0,7.2,0" BorderThickness="1,1,1,1" BorderBrush="#FF000000"><Paragraph Margin="0,0,0,0" TextAlignment="Left" FontFamily="Times New Roman" FontWeight="Bold" FontSize="14.6666666666667"><Run FontFamily="Arial" FontSize="18"></Run></Paragraph><Paragraph Margin="0,0,0,0" TextAlignment="Left" FontFamily="Times New Roman" FontSize="14.6666666666667"><Run FontFamily="Arial" FontWeight="Bold" FontSize="18" Foreground="#FF00B050">Definition: </Run><Run FontFamily="Arial" FontSize="18">A </Run><Run FontFamily="Arial" FontStyle="Italic" FontWeight="Bold" FontSize="18">geometric sequence</Run><Run FontFamily="Arial" FontSize="18"> is one in which each term is a common multiple of its preceding term. Every consecutive term differs by a </Run><Run FontFamily="Arial" FontStyle="Italic" FontSize="18">common ratio</Run><Run FontFamily="Arial" FontWeight="Bold" FontSize="18">.</Run><Run FontFamily="Arial" FontSize="18"> Geometric sequences are of the form </Run><Image Stretch="None"><Image.Source><BitmapImage UriSource="38567a5fefb94a0e9eb17858aedeb7da.png" CacheOption="OnLoad" /></Image.Source></Image><Run FontFamily="Arial" FontSize="18">, where </Run><Image Stretch="None"><Image.Source><BitmapImage UriSource="ee8071bcf12d454bba764b6f2d846941.png" CacheOption="OnLoad" /></Image.Source></Image><Run FontFamily="Arial" FontSize="18"> is the first term, </Run><Image Stretch="None"><Image.Source><BitmapImage UriSource="a9d302f10ac2432ea825612e443637be.png" CacheOption="OnLoad" /></Image.Source></Image><Run FontFamily="Arial" FontSize="18"> is the common ratio. This sequence contains </Run><Image Stretch="None"><Image.Source><BitmapImage UriSource="908bcbe043474fd8922809fcfa47f991.png" CacheOption="OnLoad" /></Image.Source></Image><Run FontFamily="Arial" FontSize="18"> terms.</Run></Paragraph><Paragraph Margin="0,0,0,0" TextAlignment="Left" FontFamily="Times New Roman" FontSize="14.6666666666667"><Run FontFamily="Arial" FontSize="18"></Run></Paragraph></TableCell></TableRow></TableRowGroup></Table><Paragraph Margin="0,0,0,0" TextAlignment="Left" FontFamily="Times New Roman" FontSize="14.6666666666667"><Run FontFamily="Arial" FontSize="18"></Run></Paragraph><Paragraph Margin="0,0,0,0" TextAlignment="Left" FontFamily="Times New Roman" FontSize="14.6666666666667"><Run FontFamily="Arial" FontSize="18">Let us write our vehicle depreciation sequence in the general form of a geometric sequence:</Run></Paragraph><Paragraph Margin="0,0,0,0" TextAlignment="Left" FontFamily="Times New Roman" FontSize="14.6666666666667"><Run FontFamily="Arial" FontSize="18"></Run></Paragraph><Paragraph Margin="0,0,0,0" TextAlignment="Left" FontFamily="Times New Roman" FontSize="14.6666666666667"><Run FontFamily="Arial"></Run><Image Stretch="None"><Image.Source><BitmapImage UriSource="8b1f04cb071f41b9a71b1117b2d0142e.png" CacheOption="OnLoad" /></Image.Source></Image><Run FontFamily="Arial" FontSize="18"></Run><LineBreak /><Run FontFamily="Arial" FontSize="18" Foreground="#FF76923C"></Run></Paragraph><Paragraph Margin="0,0,0,0" TextAlignment="Left" FontFamily="Times New Roman" FontSize="14.6666666666667"><Run FontFamily="Arial" FontSize="18">So, our sequence is </Run><Image Stretch="None"><Image.Source><BitmapImage UriSource="580783bee1c64516a19c0291bfa99f74.png" CacheOption="OnLoad" /></Image.Source></Image><Run FontFamily="Arial" FontSize="18">. We can see that this is a geometric sequence, where </Run><Image Stretch="None"><Image.Source><BitmapImage UriSource="5ed09278018a466ca5338cf4f2d77dca.png" CacheOption="OnLoad" /></Image.Source></Image></Paragraph><Paragraph Margin="0,0,0,0" TextAlignment="Left" FontFamily="Times New Roman" FontSize="14.6666666666667"><Run FontFamily="Arial" FontSize="18"></Run></Paragraph><Paragraph Margin="0,0,0,0" TextAlignment="Left" FontFamily="Times New Roman" FontSize="14.6666666666667"><Run FontFamily="Arial" FontSize="18">Since the </Run><Image Stretch="None"><Image.Source><BitmapImage UriSource="bc26ef997f96422d977e75610fd72833.png" CacheOption="OnLoad" /></Image.Source></Image><Run FontFamily="Arial" FontSize="18"> term in the sequence represents how much the car will be worth at the beginning of the </Run><Image Stretch="None"><Image.Source><BitmapImage UriSource="b8cb5972cd284550b9d038c818c4dc3c.png" CacheOption="OnLoad" /></Image.Source></Image><Run FontFamily="Arial" FontSize="18"> year you own it, we can find what it will be worth 9 years from now (at the beginning of that year)! We want to find </Run><Image Stretch="None"><Image.Source><BitmapImage UriSource="081375f031624031b495f673c13b191b.png" CacheOption="OnLoad" /></Image.Source></Image><Run FontFamily="Arial" FontSize="18">, since we valued the car at $13,000 for the first year.</Run></Paragraph><Paragraph Margin="0,0,0,0" TextAlignment="Left" FontFamily="Times New Roman" FontSize="14.6666666666667"><Run FontFamily="Arial" FontSize="18"></Run></Paragraph><BlockUIContainer TextAlignment="Justify"><Image Stretch="None"><Image.Source><BitmapImage UriSource="7538ae2614804381bb5552bd0d6de658.png" CacheOption="OnLoad" /></Image.Source></Image></BlockUIContainer><Paragraph Margin="0,0,0,0" TextAlignment="Left" FontFamily="Times New Roman" FontSize="14.6666666666667"><Run FontFamily="Arial" FontSize="18"></Run></Paragraph><Paragraph Margin="0,0,0,0" TextAlignment="Left" FontFamily="Times New Roman" FontSize="14.6666666666667"><Run FontFamily="Arial" FontSize="18">The car you are considering to buy will be worth </Run><Image Stretch="None"><Image.Source><BitmapImage UriSource="7538a3a397bf47358ddddda9a7e0aef3.png" CacheOption="OnLoad" /></Image.Source></Image><Run FontFamily="Arial" FontSize="18"> at the beginning of the 9</Run><Run FontFamily="Arial" FontSize="18" Typography.Variants="Superscript">th</Run><Run FontFamily="Arial" FontSize="18"> year you have the car. </Run></Paragraph><Paragraph Margin="0,0,0,0" TextAlignment="Right" FontFamily="Times New Roman" FontWeight="Bold" FontSize="14.6666666666667"><Run FontFamily="Arial" FontSize="18"></Run><Image Stretch="None"><Image.Source><BitmapImage UriSource="3ad461e0c0014d15862530f4a073e3c1.png" CacheOption="OnLoad" /></Image.Source></Image></Paragraph><Paragraph Margin="0,0,0,0" TextAlignment="Right" FontFamily="Times New Roman" FontWeight="Bold" FontSize="14.6666666666667"><Run FontFamily="Arial" FontSize="18"></Run></Paragraph><Paragraph Margin="0,0,0,0" TextAlignment="Left" FontFamily="Times New Roman" FontSize="14.6666666666667"><Run FontFamily="Arial" FontWeight="Bold" FontSize="18">Example #2: </Run><Run FontFamily="Arial" FontSize="18">The </Run><Image Stretch="None"><Image.Source><BitmapImage UriSource="77b10108d46147a594152cbd38d08589.png" CacheOption="OnLoad" /></Image.Source></Image><Run FontFamily="Arial" FontSize="18"> term of a geometric sequence is given in each problem below. We will identify the common ratio and find the first five terms.</Run></Paragraph><Paragraph Margin="0,0,0,0" TextAlignment="Left" FontFamily="Times New Roman" FontSize="14.6666666666667"><Run FontFamily="Arial" FontSize="18"></Run></Paragraph><Paragraph Margin="0,0,0,0" TextAlignment="Left" FontFamily="Times New Roman" FontSize="14.6666666666667"><Run FontFamily="Arial" FontSize="18">a. </Run></Paragraph><BlockUIContainer><Image Stretch="None"><Image.Source><BitmapImage UriSource="442d06d0f14d41398f1d571ce31b2326.png" CacheOption="OnLoad" /></Image.Source></Image></BlockUIContainer><Paragraph Margin="0,0,0,0" TextAlignment="Left" FontFamily="Times New Roman" FontSize="14.6666666666667"><Run FontFamily="Arial" FontSize="18">The common ratio is </Run><Image Stretch="None"><Image.Source><BitmapImage UriSource="b9ea91af368144f59ec3c652858abca5.png" CacheOption="OnLoad" /></Image.Source></Image><Run FontFamily="Arial" FontSize="18">. The first five terms are as follows:</Run></Paragraph><BlockUIContainer TextAlignment="Justify"><Image Stretch="None"><Image.Source><BitmapImage UriSource="c064f92756344ac08f4954541d301867.png" CacheOption="OnLoad" /></Image.Source></Image></BlockUIContainer><Paragraph Margin="0,0,0,0" TextAlignment="Left" FontFamily="Times New Roman" FontSize="14.6666666666667"><Run FontFamily="Arial" FontSize="18"></Run></Paragraph><Paragraph Margin="0,0,0,0" TextAlignment="Left" FontFamily="Times New Roman" FontSize="14.6666666666667"><Run FontFamily="Arial" FontSize="18">b. </Run></Paragraph><Paragraph Margin="0,0,0,0" TextAlignment="Left" FontFamily="Times New Roman" FontSize="14.6666666666667"><Run FontFamily="Arial" FontSize="18"></Run></Paragraph><BlockUIContainer><Image Stretch="None"><Image.Source><BitmapImage UriSource="1db2209dd68b425caeff671360276ba2.png" CacheOption="OnLoad" /></Image.Source></Image></BlockUIContainer><Paragraph Margin="0,0,0,0" TextAlignment="Left" FontFamily="Times New Roman" FontSize="14.6666666666667"><Run FontFamily="Arial" FontSize="18"></Run></Paragraph><Paragraph Margin="0,0,0,0" TextAlignment="Left" FontFamily="Times New Roman" FontSize="14.6666666666667"><Run FontFamily="Arial" FontSize="18">We see that the common ratio is 2! The following are the first five terms of the sequence.</Run></Paragraph><Paragraph Margin="0,0,0,0" TextAlignment="Left" FontFamily="Times New Roman" FontSize="14.6666666666667"><Run FontFamily="Arial" FontSize="18"></Run></Paragraph><BlockUIContainer TextAlignment="Justify"><Image Stretch="None"><Image.Source><BitmapImage UriSource="bf1b9d5b59df44d9babc26dba00e819d.png" CacheOption="OnLoad" /></Image.Source></Image></BlockUIContainer><Paragraph Margin="0,0,0,0" TextAlignment="Left" FontFamily="Times New Roman" FontSize="21.3333333333333"><Run FontFamily="Arial" FontSize="18"></Run></Paragraph><Paragraph Margin="0,0,0,0" TextAlignment="Left" FontFamily="Times New Roman" FontSize="14.6666666666667"><Run FontFamily="Arial" FontSize="18"></Run></Paragraph><Paragraph Margin="0,0,0,0" TextAlignment="Left" FontFamily="Times New Roman" FontSize="14.6666666666667"><Run FontFamily="Arial" FontSize="18"></Run></Paragraph><Paragraph Margin="0,0,0,0" TextAlignment="Left" FontFamily="Times New Roman" FontSize="14.6666666666667"><Run FontFamily="Arial" FontSize="18"></Run></Paragraph><Paragraph Margin="0,0,0,0" TextAlignment="Right" FontFamily="Times New Roman" FontWeight="Bold" FontSize="14.6666666666667"><Run FontFamily="Arial" FontSize="18"></Run><Image Stretch="None"><Image.Source><BitmapImage UriSource="3ef162333b804489970157fb9c3c8dcd.png" CacheOption="OnLoad" /></Image.Source></Image></Paragraph><Paragraph Margin="0,0,0,0" TextAlignment="Left" FontFamily="Times New Roman" FontWeight="Bold" FontSize="14.6666666666667"><Run FontFamily="Arial" FontSize="18"></Run></Paragraph><BlockUIContainer><Image Stretch="None"><Image.Source><BitmapImage UriSource="0630148df2694c78b50e1a11506d96a5.png" CacheOption="OnLoad" /></Image.Source></Image></BlockUIContainer><Paragraph Margin="0,0,0,0" TextAlignment="Left" FontFamily="Times New Roman" FontWeight="Bold" FontSize="14.6666666666667"><Run FontFamily="Arial" FontSize="18">Salary Raise</Run></Paragraph><Paragraph Margin="0,0,0,0" TextAlignment="Left" FontFamily="Times New Roman" FontWeight="Bold" FontSize="14.6666666666667"><Run FontFamily="Arial" FontSize="18"></Run></Paragraph><Paragraph Margin="0,0,0,0" TextAlignment="Left" FontFamily="Times New Roman" FontSize="14.6666666666667"><Run FontFamily="Arial" FontSize="18">Recall the problem earlier in this unit in which your first job out of college offered you $25,000 a year and that you would get a raise of $2650 every year? Suppose instead that your raise is not a fixed amount each year but that it is a percentage of each subsequent year’s salary? In other words, your raise would be, say, 4% each year. How much would your salary be in your 13</Run><Run FontFamily="Arial" FontSize="18" Typography.Variants="Superscript">th</Run><Run FontFamily="Arial" FontSize="18"> year of employment now?</Run></Paragraph><Paragraph Margin="0,0,0,0" TextAlignment="Left" FontFamily="Times New Roman" FontSize="14.6666666666667"><Run FontFamily="Arial" FontSize="18"></Run></Paragraph><Paragraph Margin="0,0,0,0" TextAlignment="Left" FontFamily="Times New Roman" FontSize="14.6666666666667"><Run FontFamily="Arial" FontSize="18">Since you are starting with $25,000 a year, we will let </Run><Image Stretch="None"><Image.Source><BitmapImage UriSource="160e16c0b8e34a8fb433b2d2be0a3d05.png" CacheOption="OnLoad" /></Image.Source></Image><Run FontFamily="Arial" FontSize="18"> as before Now, each year, your salary increases by a fixed rate of 4%. So, our recursive system would be as follows:</Run></Paragraph><Paragraph Margin="0,0,0,0" TextAlignment="Left" FontFamily="Times New Roman" FontSize="14.6666666666667"><Run FontFamily="Arial" FontSize="18"></Run></Paragraph><BlockUIContainer TextAlignment="Justify"><Image Stretch="None"><Image.Source><BitmapImage UriSource="987d7f0d33b747d5b1f81712304cc8b7.png" CacheOption="OnLoad" /></Image.Source></Image></BlockUIContainer><Paragraph Margin="0,0,0,0" TextAlignment="Left" FontFamily="Times New Roman" FontSize="14.6666666666667"><Run FontFamily="Arial" FontSize="18"></Run></Paragraph><Paragraph Margin="0,0,0,0" TextAlignment="Left" FontFamily="Times New Roman" FontSize="14.6666666666667"><Run FontFamily="Arial" FontSize="18">This is equivalent to:</Run></Paragraph><Paragraph Margin="0,0,0,0" TextAlignment="Left" FontFamily="Times New Roman" FontSize="18.6666666666667"><Run FontFamily="Arial" FontSize="18"></Run></Paragraph><BlockUIContainer TextAlignment="Justify"><Image Stretch="None"><Image.Source><BitmapImage UriSource="19fc5600dc33474f80f2c123448e318e.png" CacheOption="OnLoad" /></Image.Source></Image></BlockUIContainer><Paragraph Margin="0,0,0,0" TextAlignment="Left" FontFamily="Times New Roman" FontSize="14.6666666666667"><Run FontFamily="Arial" FontSize="18"></Run></Paragraph><Paragraph Margin="0,0,0,0" TextAlignment="Left" FontFamily="Times New Roman" FontSize="14.6666666666667"><Run FontFamily="Arial" FontSize="18"></Run></Paragraph><Paragraph Margin="0,0,0,0" TextAlignment="Left" FontFamily="Times New Roman" FontSize="14.6666666666667"><Run FontFamily="Arial" FontSize="18">Using our calculators in sequence mode, we find that in your 13</Run><Run FontFamily="Arial" FontSize="18" Typography.Variants="Superscript">th</Run><Run FontFamily="Arial" FontSize="18"> year, your salary will be</Run></Paragraph><Paragraph Margin="0,0,0,0" TextAlignment="Left" FontFamily="Times New Roman" FontSize="14.6666666666667"><Run FontFamily="Arial" FontSize="18"></Run></Paragraph><Paragraph Margin="0,0,0,0" TextAlignment="Center" FontFamily="Times New Roman" FontWeight="Bold" FontSize="14.6666666666667"><Run FontFamily="Arial"></Run><Image Stretch="None"><Image.Source><BitmapImage UriSource="fb6c5445e0c440d0b0d634d0733614aa.png" CacheOption="OnLoad" /></Image.Source></Image></Paragraph><Paragraph Margin="0,0,0,0" TextAlignment="Left" FontFamily="Times New Roman" FontSize="14.6666666666667"><Run FontFamily="Arial" FontSize="18"></Run></Paragraph><Paragraph Margin="0,0,0,0" TextAlignment="Left" FontFamily="Times New Roman" FontSize="14.6666666666667"><Run FontFamily="Arial" FontSize="18">Recall that your salary in the 13</Run><Run FontFamily="Arial" FontSize="18" Typography.Variants="Superscript">th</Run><Run FontFamily="Arial" FontSize="18"> year was </Run><Image Stretch="None"><Image.Source><BitmapImage UriSource="284223364e384deda913e97bc0a1a528.png" CacheOption="OnLoad" /></Image.Source></Image><Run FontFamily="Arial" FontSize="18"> when you received a raise of a fixed amount. Why do you think that is?...</Run></Paragraph><Paragraph Margin="0,0,0,0" TextAlignment="Right" FontFamily="Times New Roman" FontWeight="Bold" FontSize="14.6666666666667"><Run FontFamily="Arial" FontSize="18"></Run><Image Stretch="None"><Image.Source><BitmapImage UriSource="674bdb433ada42a3a28a1b23756b3d8d.png" CacheOption="OnLoad" /></Image.Source></Image></Paragraph><Paragraph Margin="0,0,0,0" TextAlignment="Left" FontFamily="Times New Roman" FontWeight="Bold" FontSize="14.6666666666667"><Run FontFamily="Arial" FontSize="18"></Run></Paragraph><Paragraph Margin="0,0,0,0" TextAlignment="Left" FontFamily="Times New Roman" FontWeight="Bold" FontSize="14.6666666666667"><Run FontFamily="Arial" FontSize="18">What To Do Next…</Run></Paragraph><Paragraph Margin="54,0,0,0" TextAlignment="Left" FontFamily="Times New Roman" FontSize="14.6666666666667"><Run FontFamily="Arial" FontSize="18"></Run></Paragraph><Paragraph Margin="0,0,0,0" TextAlignment="Left" FontFamily="Times New Roman" FontWeight="Bold" FontSize="14.6666666666667"><Run FontFamily="Arial" FontSize="18"></Run><Image Stretch="None"><Image.Source><BitmapImage UriSource="c0f91ee52d284cecb3f289d9bd0c1526.png" CacheOption="OnLoad" /></Image.Source></Image><Run FontFamily="Arial" FontWeight="Normal" FontSize="18"></Run><Run FontFamily="Arial" FontSize="18" Foreground="#FFFF0000">We will work on problems for this lesson during our residential weekend.</Run></Paragraph><Paragraph Margin="0,0,0,0" TextAlignment="Left" FontFamily="Times New Roman" FontSize="14.6666666666667"><Run FontFamily="Arial" FontSize="18"></Run></Paragraph><Paragraph Margin="0,0,0,0" TextAlignment="Left" FontFamily="Times New Roman" FontWeight="Bold" FontSize="14.6666666666667"><Run FontFamily="Arial" FontSize="18"></Run></Paragraph></FlowDocument>