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4.6 Arithmetic Sequences How can you use an arithmetic sequence to describe a pattern? 20 ,2 5, 30 ,3 5 .. . Figure # # of circles 30 25 20 15 10 5 Patterns definition: a pattern is any group of objects or numbers that follow a rule. 1, 4, 16, 64... 3, 6, 9, 12... 2, 6, 4, 8, 6... Sequences 2, 4, 6... definition: an ordered set of numbers that follows a rule. 1, 2, 3, .... 20, 25, 30, 35... 100, 10, 1, .1,... 3, 9, 27, 81... 1, 1, 2, 3, 5, 8... There are two types of sequences we'll look at this year: arithmetic and geometric. Arithmetic Sequences definition: a sequence created by adding the same number repeatedly. Examples 5, 7, 9, 11,... +2 +2 7, 3, -1, -5,... -4 You try: +2 the number 2 is added to create this sequence. Here 2 is called the common difference. -4 -4 In this example we are subtracting 4 or adding a -4, so we say the common difference is -4 -3, 2, 7, 12,... What is the common difference? Geometric Sequences definition: a sequence created by multiplying the same number repeatedly. Examples: 1, 3, 9, 27,... x3 x3 8, 4, 2, 1,... ÷2 You try: x3 the number 3 is multiplied to each term to create this sequence. Here 3 is called the common ratio. ÷2 ÷2 In this example we are dividing by 2 or multiplying by 1/2, so we say the common ratio is 1/2 -3, -6, -12,... What is the common ratio? ers below. nd a common added to d term, and ...etc. ue to find ms. Common Difference Hint: Erase to Reveal Solution common difference the common difference between two consecutive elements of a sequence 1. 7, 11, 15, 19, 23, 27, 31 2. -4, -6, -8, -10, -12, -14, -16 3. 15, 7, -1, -9, -17, -25, -33 Given First Term and Common Difference Hint: Move to Reveal Solution a1 = the first term of an arithmetic sequence d = the common difference (value add to each term to find the next term) Find the first five terms of each sequence. Check your solution by moving the question to the solution box. Question 1. a1= 6 d = 9 6, 15, 24, 33, 42 2. a1 = -60 d = 4 -60, -56, -52, -48, -44 Solution Write the next three terms of the arithmetic sequence. 1. −12, 0, 12, 24, . . . 2. 0.2, 0.6, 1, 1.4, . . . 3. Does the graph represent an arithmetic sequence? Explain. an - value of the nth number in the sequence a1 - first number in sequence n - step number d - common difference an= a1 + (n - 1)d Write an equation for the nth term of the arithmetic sequence 14, 11, 8, 5, . . .. Then find a50. an= a1 + (n - 1)d Don't Forget! (n - 1)d Write an equation for the nth term of the arithmetic sequence. Then find a25. a. 4, 5, 6, 7, . . . b. 8, 16, 24, 32, . . . c. 1, 0, −1, −2, . . . Don't Forget! Online bidding for a purse Bid Number 1 increases by $5 for each bid Bid Amount $60 $65 $70 $75 2 3 after the $60 initial bid. a. Write a function that represents the arithmetic sequence. b. Graph the function. c. The winning bid is $105. How many bids were there? 4 A carnival charges $2 for each game after you pay a $5 entry fee. a. Write a function that represents the arithmetic sequence. b. Graph the function. c. How many games can you play when you take $29 to the carnival? Games 1 2 3 4 Total Cost $7 $9 $11 $13 Pg 214: 1-15odd, 19, 24-36e, 46-48, 62-65 3-2-1: Hand out a reflection sheet as described in the Formative Assessment Tips. Use the table to find the slope. 1. x 2 3 4 5 2. x 14 28 42 56 70 y 17 20 23 26 29 y 4 11 18 25 3. x 5 21 37 53 4. x 0 4 8 27 43 49 y 4 8 12 16 20 y 11 12 16 Tell which number you would add to or subtract from each side of the inequality to solve. 1. k − 12 > −4 2. 0 ≤ b + 8 3. x + 5 > −6 4. 7 ≤ m + 2 5. r − 2 > 6 6. 8 + w > 8 Work with a partner. Use the figures to complete the table. Plot the points given by your completed table. Describe the pattern of the y-values. Graph the arithmetic sequence 4, 8, 12, 16, . . .. What do you notice? Graph the arithmetic sequence. What do you notice? 4. 3, 6, 9, 12, . . . 5. 4, 2, 0, −2, . . . 6. 1, 0.8, 0.6, 0.4, . . . 7. Does the graph shown represent an arithmetic sequence? Explain.