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5 ∑ = 1 3 20 ∑ = 1 12 + 12 20 2 Enriching the Lesson Tell students the following legend: When the famous mathematician Leonhard Euler was in grade school, the school master, in an attempt to keep his students busy, told the students to find the sum of the natural numbers from 1 to 100. The school master was quite disappointed when Leonhard was able to produce the correct answer in about sixty seconds. Show students the following method that was used by Euler. Then show students how the formula S = n ( a + a ) is a generalization of that method. n 2 1 1 2 3 . . . . 98 99 100 + 100 + 99 + 98 . . . . + 3 + 2 + 1 101 101 101 101 101 101 101 101 101 101 Sum = 101(100) ÷ 2 = 5,050 n INSTRUCTION Help students to learn summation notation by translating the symbols into English. For example, 5 ∑ n= 1 3n is read as “the sum of the values of 3n from n = 1 to n = 5.” Have students revisit the handshake problem which was investigated in Lesson 9-1. Ask students to write a summation notation to represent the number of handshakes. Make sure that students do not assume that the upper limit of a summation notation is equivalent to the number of terms that are being summed. Advise students that the index of a summation notation is not always 1. Make sure that students realize that the formula S n n = ( a + a n) 2 1 does not apply to a summation 7 2 notation such as: ∑ n . n= 1 Example 3 Reinforce to students that the formula S n n = ( a + a n) is just a 2 1 generalization of Euler’s method which requires less work. Show students how his method could be used to determine the number of seats in the auditorium. 9.2 Arithmetic Sequences and Series 399
INSTRUCTION Ongoing Assessment Show students how the graph of the equation y = 4(x + 1) would show all of the terms in this series. Emphasize that only the points with x-values that are natural numbers are part of the series. WRAP UP To ensure mastery of objectives, students should be able to: • Determine if a given sequence is arithmetic and state the common difference. • Use the arithmetic mean to determine missing terms in an arithmetic sequence. • Evaluate a given summation notation for an arithmetic series. Assignment In-class practice: 1–5 Homework: 6–34 Math Applications Exercises 5, 9, 11, and 12 from pages 422–429 4( + 1) = 1 16 ∑ 1 , 2 , 3 3 1 , 4 , 5 ,… 1 3 3 3 Think and Discuss Answers 1. Find the difference between any two consecutive terms of the sequence. 2. Answers will vary. Sample answer: Use the formula S n n = ( a + a n) where 2 a 1 1 = 1, a n = 100, and n = 100. 3. An arithmetic sequence is a pattern of numbers while an arithmetic series is the sum of the numbers in the sequence. 4. Answers will vary. Sample answer: the arithmetic mean is the average of the two numbers. 5. yes; the sequence 10, 8, 6, 4, 2, … has a common difference of –2. 400 Chapter 9 Sequences and Series
Page 1 and 2: LESSON OBJECTIVES 9.1 Patterns •
Page 3 and 4: LESSON PLANNING Vocabulary sequence
Page 5 and 6: INSTRUCTION Example 2 Return to the
Page 7 and 8: Answers to Math Applications Math A
Page 9: INSTRUCTION Tell students that the
Page 15 and 16: INSTRUCTION Tell students that the
Page 17 and 18: Problem Solving Before solving this
Page 21 and 22: INSTRUCTION Students might incorrec
Page 23 and 24: WRAP UP To ensure mastery of object
Page 25 and 26: Mixed Review Additional Answers 35.
Page 27 and 28: INSTRUCTION Ask students to express
Page 29 and 30: WRAP UP To ensure mastery of object
Page 31 and 32: MATH LAB Activity 1 PREPARE • An
Page 33 and 34: Math Applications Solutions and Not
Page 41 and 42: Vocabulary Review arithmetic mean (
Page 43 and 44: 1 2 10 , 922 1 2 ∞ 1 ∑

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