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3-6

3-6. Arithmetic Sequences. Course 3. Warm Up. Problem of the Day. Lesson Presentation. Warm Up Use the table to write an equation. 1. 2. 3. y = 5 x. y = –2.5 x. y = 3 x + 2. Problem of the Day

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3-6

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  1. 3-6 Arithmetic Sequences Course 3 Warm Up Problem of the Day Lesson Presentation

  2. Warm Up Use the table to write an equation. 1. 2. 3. y = 5x y = –2.5x y = 3x + 2

  3. Problem of the Day A movie ticket at a certain theater costs $4.50 for a child and $8.75 for an adult. How much will it cost for a family of two adults and three children to see a movie? $31

  4. Learn to identify and evaluate arithmetic sequences.

  5. Vocabulary sequence term arithmetic sequence common difference

  6. A sequence is an ordered list of numbers or objects, called terms. In an arithmetic sequence, the difference between one term and the next is always the same. This difference is called the common difference. The common difference is added to each term to get the next term.

  7. Additional Example 1: Finding the Common Difference in an Arithmetic Sequence Find the common difference in each arithmetic sequence. A. 11, 9, 7, 5, … 11, 9, 7, 5, … The terms decrease by 2. –2 –2 –2 The common difference is -2. B. 2.5, 3.75, 5, 6.25, … 2.5, 3.75, 5, 6.25, … The terms increase by 1.25. +1.25 +1.25 +1.25 The common difference is 1.25.

  8. Check It Out: Example 1 Find the common difference in each arithmetic sequence. A. 2, 6, 10, 14, … 2, 6, 10, 14, … The terms increase by 4. +4 +4 +4 The common difference is 4. B. 2.5, 5, 7.5, 10, … 2.5, 5, 7.5, 10, … The terms increase by 2.5. +2.5 +2.5 +2.5 The common difference is 2.5.

  9. Additional Example 2: Finding Missing Terms in an Arithmetic Sequence Find the next three terms in the arithmetic sequence –8, –3, 2, 7, ... Each term is 5 more than the previous term. 7 + 5 = 12 Use the common difference to find the next three terms. 12 + 5 = 17 17 + 5 = 22 The next three terms are 12, 17, and 22.

  10. Check It Out: Example 2 Find the next three terms in the arithmetic sequence –9, –6, –3, 0, ... Each term is 3 more than the previous term. 0 + 3 = 3 Use the common difference to find the next three terms. 3 + 3 = 6 6 + 3 = 9 The next three terms are 3, 6, and 9.

  11. Additional Example 3A: Identifying Functions in Arithmetic Sequences Find a function that describes each arithmetic sequence. 6, 12, 18, 24, … 1• 6 6 Multiply n by 6. 2• 6 12 3• 6 18 y = 6n 4• 6 24 n• 6 6n

  12. Additional Example 3B: Identifying Functions in Arithmetic Sequences Find a function that describes each arithmetic sequence. –4, –8, –12, –16, … 1• (–4) –4 Multiply n by -4. 2• (–4) –8 y = –4n 3• (–4) –12 4• (–4) –16 n• (–4) –4n

  13. Check It Out: Example 3A Find a function that describes each arithmetic sequence. 3, 6, 9, 12, … 1• 3 3 Multiply n by 3. 2• 3 6 3• 3 9 y = 3n 4• 3 12 n• 3 3n

  14. Check It Out: Example 3B Find a function that describes each arithmetic sequence. –7, –14, –21, –28, … 1• (-7) -7 Multiply n by -7. 2• (-7) -14 y = -7n 3• (-7) -21 4• (-7) -28 n• (-7) -7n

  15. Additional Example 4: Application A DVD costs $3.95 to rent, plus $0.45 for each day it is returned late. Find a function that describes the arithmetic sequence, and then find the total cost of renting the DVD and returning it 9 days late. Multiply n by $0.45, and then add the $3.95 rental fee. 3.95 + 0.45(1) 4.40 3.95 + 0.45(2) 4.85 3.95 + 0.45(3) 5.30 3.95 + 0.45(4) 5.75 3.95 + 0.45(5) 0.45n + 3.95

  16. Additional Example 4 Continued A DVD costs $3.95 to rent, plus $0.45 for each day it is returned late. Find a function that describes the arithmetic sequence, and then find the total cost of renting the DVD and returning it 9 days late. 0.45n + 3.95 Write a function to find the 9th term. 0.45(9) + 3.95 Substitute 9 for n. 4.05 + 3.95 Multiply. 8.00 Add. It will cost $8.00 to rent a movie and return it 9 days late.

  17. Check It Out: Example 4 A personal pizza with cheese costs $3.99 to buy, plus $0.25 for each additional topping. Find a function that describes the arithmetic sequence, and then find the total cost of buying a personal pizza with 7 additional toppings. Multiply n by $0.25, and then add the $3.99 pizza cost. 3.99 + 0.25(1) 4.24 3.99 + 0.25(2) 4.49 3.99 + 0.25(3) 4.74 3.99 + 0.25(4) 4.99 3.99 + 0.25(5) 0.25n + 3.99

  18. Check It Out: Example 4 Continued A personal pizza with cheese costs $3.99 to buy, plus $0.25 for each additional topping. Find a function that describes the arithmetic sequence, and then find the total cost of buying a personal pizza with 7 additional toppings. 0.25n + 3.99 Write a function to find the 7th term. 0.25(7) + 3.99 Substitute 7 for n. 1.75 + 3.99 Multiply. 5.74 Add. It will cost $5.74 to buy a personal pizza with 7 additional toppings.

  19. Lesson Quiz: Part 1 Find the common difference in each arithmetic sequence. 1. 4, 2, 0, –2, … –2 8 3 4 3 10 3 2 3 2. , 2, , , … Find the next three terms in each arithmetic sequence. –2, –7, –12 3. 18, 13, 8, 3, … 4. 3.6, 5, 6.4, 7.8, … 9.2, 10.6, 12

  20. Lesson Quiz: Part 2 Find a function that describes the arithmetic sequence. 5. –5, –10, –15, –20, … Possible Answer: y = –5n 6. –1, 2, 5, 8, … Possible Answer: y = 3n – 4 7. A runner finishes a lap in 55 seconds. Her goal is to decrease her time by two seconds every week. Find a function that describes the arithmetic sequence and find how many seconds her lap should be after 12 weeks of training. Possible Answer: s = -2w + 55; 31 seconds

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