Arithmetic Series. or The sum of any finite, or limited, number of terms is called a partial sum. are shorthand ways of writing n 1


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1 CONDENSED LESSON 9.1 Arithmetic Series In this lesson you will learn the terminology and notation associated with series discover a formula for the partial sum of an arithmetic series A series is the indicated sum of terms of a sequence. For example, consider the sequence 4 u n u n1 2 where n 2 The sum of the terms in this sequence is the series u 2 u 3 u 4 or The sum of the first n terms in a series is represented by. For example, S 6 u 2 u 3 u 4 u 5 u The sum of any finite, or limited, number of terms is called a partial sum 6 of the series. The notations S 6 and u n are shorthand ways of writing n 1 u 2 u 3 u 4 u 5 u 6. To find the sum of the integers from 1 to 100, you could add the terms one by one. You can use technology and a recursive formula to do this quickly. First, write a recursive definition for the sequence of positive integers. Sequence: 1 u n u n1 1 where n 2 Then, write the definition for the related series. Remember, the sum of the first 100 terms is the sum of the first 99 terms plus the 100th term. Series: S u n where n 2 The table shows each term in the sequence and the sequence of partial sums. The points on the graph represent the partial sums S 1 through S 100. You can use either the table or the graph to find that S 100, the sum of the integers from 1 to 100, is (continued) Discovering Advanced Algebra Condensed Lessons CHAPTER Key Curriculum Press
2 Lesson 9.1 Arithmetic Series (continued) In the investigation you will find a formula for finding a partial sum of an arithmetic series without finding all the terms and adding. Investigation: Arithmetic Series Formula Work through Steps 1 and 2 of the investigation in your book. If you have the materials, complete the rest of the investigation. Then check your work against the solution below. Step 1 The length of the first step is 4, the second is 7, and so on until the last step, which is 16. Sequence: 4, 7, 10, 13, 16 Sum of the series: Step 2 The dimensions of the rectangle are 20 units by 5 units. Note that the area is 100 square units, twice the value of the sum of the series. u 2 u 3 u 4 u 5 Slide Steps 3 and 4 Use the sequence 2, 4, 6, 8. Then 2, d 2. Note that the related series is The figure below shows two copies of a stepshaped figure representing the sequence. The dimensions of the rectangle are 10 units by 4 units, giving an area of 40 square units. u 2 u 3 u 4 Slide The area of the rectangle is given by n u 4. The length of the rectangle is equal to the sum of the first and last terms of the sequence, u 4, and the height of the rectangle is equal to n, the number of terms in the sequence. Step 5 The partial sum,, of an arithmetic series is n u n 2. This is onehalf of the area of the rectangle. Use the formula from the investigation to verify that the sum of the integers from 1 to 100 is Then read the example in your book and the text following it. 138 CHAPTER 9 Discovering Advanced Algebra Condensed Lessons
3 CONDENSED LESSON 9.2 Infinite Geometric Series In this lesson you will learn that some infinite geometric series converge to a longrun value, or sum discover a formula for finding the sum of a convergent geometric series In Lesson 9.1, you found partial sums of arithmetic series. If you start adding terms of an arithmetic sequence, the magnitude of the partial sum increases. This eventually happens even if the terms are small, as in 0.001, 0.002, 0.003, and so on. This is not always the case with a geometric series. A geometric series is the summation of terms in a geometric sequence. For example, consider the geometric sequence 1 2, 1, 1 4 8, 1 16, 1 32, 1 64, 1 128,... This series has a constant ratio of 1, 2 so the terms get smaller and smaller. You can add the terms to create a geometric series. Here are some of the partial sums: S S S If you continue to find partial sums, you will get , , 128, and so on. Although the partial sums get larger and larger, they are always less than 1. It appears that if you add an infinite number of terms, the result will not be infinite. An infinite geometric series is a geometric series with an infinite number of terms. A convergent series is a series for which the sequence of partial sums approaches a longrun value as the number of terms increases. This longrun value is the sum of the series. The series 1_ 2 1_ 4 1_ is a convergent series with a longrun value, or sum, of 1. Work through Example A in your book. Investigation: Infinite Geometric Series Formula Work through the investigation yourself before reading the solutions below. Step 1 The first term,, is 0.4. The common ratio, r, is , or 0.1. The multiplier and r are reciprocals. You could use any power of ten as a multiplier. Step 2 Let S Then 0.1S = Subtract S and 0.1S and then solve for S. S S S 0.4 S , or 4 9 This method still resulted in S 4 9. (continued) Discovering Advanced Algebra Condensed Lessons CHAPTER 9 139
4 Lesson 9.2 Infinite Geometric Series (continued) Step 3 The first term,, is 0.9. The ratio, r, is 0.1. Let S and 0.1S Subtract S and 0.1S and then solve for S. S S S 0.9 S 1 Step 4 The first term,, is The ratio, r, is Let S and 0.01S Subtract and then solve for S. S S S 0.27 S Step 5 If S r r 2 r 3..., then r S r r r 2 r 3..., or r r 2 r Subtract and solve for S. S r + r 2 r 3... r S r r 2 r 3... S rs S (1 r) u S 1 1 r Subtract. Factor. Divide both sides by (1 r). Step 6 The partial sums of a geometric sequence will converge to a unique number S when r is between 1 and 1, or when 0. Read Example B in your book, in which a graph of partial sums is used to find the sum of a series. Read the example carefully and make sure you understand the method. Then, read the box after that example, which summarizes the formula for finding the sum of a convergent infinite geometric series. Note that a geometric series converges only if r 1 or 0. Then work through Example C. Here is another example. EXAMPLE Solution Find the sum of the infinite series n1 130(0.84) n1 In this case, r 0.84 and 130. Using the formula S S r, 140 CHAPTER 9 Discovering Advanced Algebra Condensed Lessons
5 CONDENSED LESSON 9.3 Partial Sums of Geometric Series In this lesson you will discover a formula for partial sums of geometric series In Lesson 9.2, you found sums of convergent geometric series. In this lesson, you will find partial sums of geometric series. Example A in your book shows you how to use a calculator table or graph to find partial sums of a geometric series. Read the example carefully. In Lesson 9.1, you discovered a formula for partial sums of arithmetic series. In this investigation, you ll find a formula for partial sums of geometric series. Investigation: Geometric Series Formula Work through the investigation in your book. Then check your work against the results below. Step 1 The sequence is defined by 180 and u n 0.65 u n1. The first ten heights and partial sum are given in the tables below. Step 2 The scatterplot of the data is shown below. The longrun value L is given by 1 r To find the values of a and b, substitute the coordinates of the points (1, 180) and (2, 297) into 180 ab n to get the system ab ab 2 You can rewrite these equations as ab and ab Dividing the second equation by the first gives b Substituting 0.65 for b in the first equation gives 0.65a So a So the equation is (0.65)n, or as an exponential function, y (0.65)x. (continued) Discovering Advanced Algebra Condensed Lessons CHAPTER 9 141
6 Lesson 9.3 Partial Sums of Geometric Series (continued) u Step 3 The equation from Step 2 can be rewritten as 1 (1 r 1 r n ) Factor out 1 r n 1 r Step 4 1 r. Rewrite the equation. r + r 2... r n1 r r r 2... r n1 r n r r n, or 1 r n (1 r) 1 r n 1 r n 1 r Step 5 S 10 for the bouncing ball is given by S r This can be verified on the calculator table. For the geometric sequence 2, 6, 18, 54, and so on, 2 and r 3. S , r r n. Now you have an explicit formula for finding a partial sum of any geometric series. You need to know only the first term, the common ratio, and the number of terms. To practice using the formula, work through Examples B and C in your book. Then read the example below. EXAMPLE Find each partial sum. 11 a. 9(2.75) n1 n1 b Solution a. 9 and r Use the formula for the partial sum S 11. S 11 1 r 11 (1 r) , b. The first term,, is Each term is threefourths the previous term, so r Enter 1024 and u n 0.75u n1 into your calculator and make a table. The last term given, , is u 8. So you need to find S 8. Using the formula, S 8 1 r 8 (1 r) CHAPTER 9 Discovering Advanced Algebra Condensed Lessons
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