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**Terms of Arithmetic Sequences**13-1 Course 3 Warm Up Problem of the Day Lesson Presentation**Terms of Arithmetic Sequences**13-1 Course 3 Warm Up Find the next two numbers in the pattern, using the simplest rule you can find. 1.1, 5, 9, 13, . . . 2. 100, 50, 25, 12.5, . . . 3. 80, 87, 94, 101, . . . 4. 3, 9, 7, 13, 11, . . . 17, 21 6.25, 3.125 108, 115 17, 15**Problem of the Day**Write the last part of this set of equations so that its graph is the letter W. y = –2x + 4 for 0 x 2 y = 2x – 4 for 2 < x 4 y = –2x + 12 for 4 < x 6 Possible answer: y = 2x – 12 for 6 < x 8**Caution!**You cannot tell if a sequence is arithmetic by looking at a finite number of terms because the next term might not fit the pattern.**Additional Example 1A: Identifying Arithmetic Sequences**Determine if the sequence could be arithmetic. If so, give the common difference. 5, 8, 11, 14, 17, . . . The terms increase by 3. 5 8 11 14 17, . . . 3 3 3 3 The sequence could be arithmetic with a common difference of 3.**Additional Example 1B: Identifying Arithmetic Sequences**Determine if the sequence could be arithmetic. If so, give the common difference. 1, 3, 6, 10, 15, . . . Find the difference of each term and the term before it. 1 3 6 10 15, . . . 5 4 2 3 The sequence is not arithmetic.**Additional Example 1C: Identifying Arithmetic Sequences**Determine if the sequence could be arithmetic. If so, give the common difference. 65, 60, 55, 50, 45, . . . The terms decrease by 5. 65 60 55 50 45, . . . –5 –5 –5 –5 The sequence could be arithmetic with a common difference of –5.**Additional Example 1D: Identifying Arithmetic Sequences**Determine if the sequence could be arithmetic. If so, give the common difference. 5.7, 5.8, 5.9, 6, 6.1, . . . The terms increase by 0.1. 5.7 5.8 5.9 6 6.1, . . . 0.1 0.1 0.1 0.1 The sequence could be arithmetic with a common difference of 0.1.**Additional Example 1E: Identifying Arithmetic Sequences**Determine if the sequence could be arithmetic. If so, give the common difference. 1, 0, -1, 0, 1, . . . Find the difference of each term and the term before it. 1 0 –1 0 1, . . . 1 1 –1 –1 The sequence is not arithmetic.**Check It Out: Example 1A**Determine if the sequence could be arithmetic. If so, give the common difference. 1, 2, 3, 4, 5, . . . The terms increase by 1. 1 2 3 4 5, . . . 1 1 1 1 The sequence could be arithmetic with a common difference of 1.**Check It Out: Example 1B**Determine if the sequence could be arithmetic. If so, give the common difference. 1, 3, 7, 8, 12, … Find the difference of each term and the term before it. 1 3 7 8 12, . . . 4 1 2 4 The sequence is not arithmetic.**Check It Out: Example 1C**Determine if the sequence could be arithmetic. If so, give the common difference. 11, 22, 33, 44, 55, . . . The terms increase by 11. 11 22 33 44 55, . . . 11 11 11 11 The sequence could be arithmetic with a common difference of 11.**Check It Out: Example 1D**Determine if the sequence could be arithmetic. If so, give the common difference. 1, 1, 1, 1, 1, 1, . . . Find the difference of each term and the term before it. 1 1 1 1 1, . . . 0 0 0 0 The sequence could be arithmetic with a common difference of 0.**Check It Out: Example 1E**Determine if the sequence could be arithmetic. If so, give the common difference. 2, 4, 6, 8, 9, . . . Find the difference of each term and the term before it. 2 4 6 8 9, . . . 1 2 2 2 The sequence is not arithmetic.**Helpful Hint**Subscripts are used to show the positions of terms in the sequence. The first term is a1, “read a sub one,” the second is a2, and so on.**Additional Example 2A: Finding a Given Term of an Arithmetic**Sequence Find the given term in the arithmetic sequence. 10th term: 1, 3, 5, 7, . . . an = a1 + (n – 1)d a10 = 1 + (10 – 1)2 a10 = 19**Additional Example 2B: Finding a Given Term of an Arithmetic**Sequence Find the given term in the arithmetic sequence. 18th term: 100, 93, 86, 79, . . . an = a1 + (n – 1)d a18 = 100 + (18 – 1)(–7) a18 = -19**Additional Example 2C: Finding a Given Term of an Arithmetic**Sequence Find the given term in the arithmetic sequence. 21st term: 25, 25.5, 26, 26.5, . . . an = a1 + (n – 1)d a21 = 25 + (21 – 1)(0.5) a21 = 35**Additional Example 2D: Finding a Given Term of an Arithmetic**Sequence Find the given term in the arithmetic sequence. 14th term: a1 = 13, d = 5 an = a1 + (n – 1)d a14 = 13 + (14 – 1)5 a14 = 78**Check it Out: Example 2A**Find the given term in the arithmetic sequence. 15th term: 1, 3, 5, 7, . . . an = a1 + (n – 1)d a15 = 1 + (15 – 1)2 a15 = 29**Check It Out: Example 2B**Find the given term in the arithmetic sequence. 50th term: 100, 93, 86, 79, . . . an = a1 + (n – 1)d a50 = 100 + (50 – 1)(-7) a50 = –243**Check It Out: Example 2C**Find the given term in the arithmetic sequence. 41st term: 25, 25.5, 26, 26.5, . . . an = a1 + (n – 1)d a41 = 25 + (41 – 1)(0.5) a41 = 45**Check It Out: Example 2D**Find the given term in the arithmetic sequence. 2nd term: a1 = 13, d = 5 an = a1 + (n – 1)d a2 = 13 + (2 – 1)5 a2 = 18**You can use the formula for the nth term of an arithmetic**sequence to solve for other variables.**Additional Example 3: Application**The senior class held a bake sale. At the beginning of the sale, there was $20 in the cash box. Each item in the sale cost 50 cents. At the end of the sale, there was $63.50 in the cash box. How many items were sold during the bake sale? Identify the arithmetic sequence: 20.5, 21, 21.5, 22, . . . a1 = 20.5 a1 = 20.5 = money after first sale d = 0.5 d = .50 = common difference an = 63.5 an = 63.5 = money at the end of the sale**Additional Example 3 Continued**Let n represent the item number of cookies sold that will earn the class a total of $63.50. Use the formula for arithmetic sequences. an = a1 + (n – 1) d 63.5 = 20.5 + (n – 1)(0.5) Solve for n. Distributive Property. 63.5 = 20.5 + 0.5n – 0.5 63.5 = 20 + 0.5n Combine like terms. 43.5 = 0.5n Subtract 20 from both sides. Divide both sides by 0.5. 87 = n During the bake sale, 87 items are sold in order for the cash box to contain $63.50.**Check It Out: Example 3**Johnnie is selling pencils for student council. At the beginning of the day, there was $10 in his money bag. Each pencil costs 25 cents. At the end of the day, he had $40 in his money bag. How many pencils were sold during the day? Identify the arithmetic sequence: 10.25, 10.5, 10.75, 11, … a1 = 10.25 a1 = 10.25 = money after first sale d = 0.25 d = .25 = common difference an = 40 an = 40 = money at the end of the sale**Check It Out: Example 3 Continued**Let n represent the number of pencils in which he will have $40 in his money bag. Use the formula for arithmetic sequences. an = a1 + (n – 1)d 40 = 10.25 + (n – 1)(0.25) Solve for n. 40 = 10.25 + 0.25n – 0.25 Distributive Property. Combine like terms. 40 = 10 + 0.25n Subtract 10 from both sides. 30 = 0.25n 120 = n Divide both sides by 0.25. 120 pencils are sold in order for his money bag to contain $40.**27**5 3 7 , or 6.75 4 4 2 4 Lesson Quiz Determine if each sequence could be arithmetic. If so, give the common difference. 1. 42, 49, 56, 63, 70, . . . 2. 1, 2, 4, 8, 16, 32, . . . Find the given term in each arithmetic sequence. 3. 15th term: a1 = 7, d = 5 4. 24th term: 1, , , , 2 5. 52nd term: a1 = 14.2; d = –1.2 yes; 7 no 77 –47