12.1/12.2 – Arithmetic and Geometric Sequences

Sequences Ordered list of numbers Each number in the list is called a term of the sequence

Arithmetic Sequences Recursive definition: Explicit definition: An arithmetic sequence is a sequence for which consecutive terms have a common difference, d. Ex: 0, 4, 8, 12, 16, 20, 24,… has a common difference of ____ Recursive definition: Relates one term to its previous term 𝑎 1 =𝑎 and 𝑎 𝑛 = 𝑎 𝑛−1 +𝑑 Explicit definition: 𝑎 𝑛 = 𝑎 1 +𝑑 𝑛−1

Ex: What is the 100th term of the arithmetic sequence that begins 6,11,…?

Ex: Use the given table to write an explicit and a recursive rule for the sequence. First, check the differences of consecutive values: 8 - 5 = 3 11 - 8 = 3 14 - 11 = 3 17 - 14 = 3 The differences are the same, so the sequence is arithmetic. The initial term 𝑎 1 of the sequence is 2. As already observed, the common difference d is 3. So the recursive rule is: 𝑎 1 =2 and 𝑎 𝑛 = 𝑎 𝑛−1 +3 The explicit rule is: 𝑎 𝑛 =2+3 𝑛−1

Ex: Use the given table to write an explicit and a recursive rule for the sequence. First, check the differences of consecutive values: 25 - 29 = -4 21 - 25 = -4 17 - 21 = -4 13 - 17 = -4 9 - 13 = -4 The differences are the same, so the sequence is arithmetic. The initial term 𝑎 1 of the sequence is 29. As already observed, the common difference d is -4. So the recursive rule is: 𝑎 1 =29 and 𝑎 𝑛 = 𝑎 𝑛−1 −4 The explicit rule is: 𝑎 𝑛 =29−4 𝑛−1

On Your Own: Use the given table to write an explicit and a recursive rule for the sequence. First, check the differences of consecutive values: 5 - 11 = -6 -1 - 5 = -6 -7 – (-1) = -6 -7 – (-13) = -6 -19 – (-13) = -6 The differences are the same, so the sequence is arithmetic. The initial term 𝑎 1 of the sequence is 11. As already observed, the common difference d is -6. So the recursive rule is: 𝑎 1 =11 and 𝑎 𝑛 = 𝑎 𝑛−1 −6 The explicit rule is: 𝑎 𝑛 =11−6 𝑛−1

Graphing Arithmetic Sequences The arithmetic sequence 3, 7, 11, 15, 19 has a final term, so it is called a finite sequence and its graph has a countable number of points. The arithmetic sequence 3, 7, 11, 15, 19, … does not have a final term (indicated by the three dots), so it is called an infinite sequence and its graph has infinitely many points. Note: Since you cannot show the complete graph of an infinite sequence, you should simply show as many points as the grid allows.

Ex: Write the terms of the given arithmetic sequence and then graph the sequence. 𝑎 𝑛 =−1+2𝑛 for 0≤𝑛≤4

Ex: Write the terms of the given arithmetic sequence and then graph the sequence. 𝑎 1 =−3 and 𝑎 𝑛 = 𝑎 𝑛−1 −1 for 𝑛≥1

Ex: Write a recursive rule and an explicit rule for an arithmetic sequence that models the situation. Then use the rule to answer the question. There are 19 seats in the row nearest the stage of a theater. Each row after the first one has 2 more seats than the row before it. How many seats are in the 13th row? Let n represent the row number, starting with 1 for the first row. The verbal description gives you a recursive rule: 𝑎 1 =19 and 𝑎 𝑛 = 𝑎 𝑛−1 + 2 for 𝑛≥2. Since the initial term is 19 and the common difference is 2, an explicit rule is 𝑎 𝑛 =19+2(𝑛−1) for 𝑛≥1. To find the number of seats in the 13th row, use the explicit rule. 𝑎 13 =19+2 13−1 =19+2 12 =19+24=43 So, there are 43 seats in the 13th row.

Ex: Write a recursive rule and an explicit rule for an arithmetic sequence that models the situation. Then use the rule to answer the question. The starting salary for a summer camp counselor is $395 per week. In each of the subsequent weeks, the salary increases by $45 to encourage experienced counselors to work for the entire summer. If the salary is $710 for the last week of the camp, for how many weeks does the camp run? Let n represent the week number, starting with 1 for the first week. The verbal description gives you a recursive rule: 𝑎 1 =395 and 𝑎 𝑛 = 𝑎 𝑛−1 +45 for 𝑛≥2. Since the initial term is 395 and the common difference is 45, an explicit rule is 𝑎 𝑛 =395+45(𝑛−1) for 𝑛≥1. To determine how long the camp runs, find the value of n for which 𝑎 𝑛 =710. Using the explicit rule: 𝑎 𝑛 =395+45 𝑛−1 710=395+45 𝑛−1 315=45(𝑛−1) 7=𝑛−1 𝑛=8 So, the summer camp runs for 8 weeks.

Geometric Sequences A geometric sequence is a sequence where the ratio between consecutive terms is constant. This ratio is the common ratio, r. It has the form: 𝑎,𝑎𝑟,𝑎 𝑟 2 ,𝑎 𝑟 3 ,…

Ex: Is the sequence geometric? If it is, what are 𝑎 1 and 𝑟? a) 3,6,12,24,48,… Yes because it has a common ratio of 2 𝑎 1 =3 and 𝑟=2 b) 3,6,9,12,15,… No because it does not have a common ratio

Recursive and Explicit Definitions for Geometric Sequences Recursive definition: Relates one term to its previous term 𝑎 1 =𝑎 and 𝑎 𝑛 =𝑟∙ 𝑎 𝑛−1 Explicit definition: 𝑎 𝑛 = 𝑎 1 ∙ 𝑟 𝑛−1

Ex: What is the 10th term of the geometric sequence 4,12,36,…?

Ex: Use the given table to write an explicit and a recursive rule for the sequence. First, check the ratios of consecutive values: 1 5 1 25 = 25 5 =5 1 1 5 = 5 1 =5 5 1 =5 25 5 =5 The ratios are the same, so the sequence is geometric. The initial term 𝑎 1 of the sequence is 1/25. As already observed, the common ratio r is 5. So the recursive rule is: 𝑎 1 = 1 25 and 𝑎 𝑛 = 𝑎 𝑛−1 ∙5 The explicit rule is: 𝑎 𝑛 = 1 25 ∙ (5) 𝑛−1

On Your Own: Use the given table to write an explicit and a recursive rule for the sequence. First, check the ratios of consecutive values: 0.01 0.001 =10 0.1 0.01 =10 1 0.1 =10 10 1 =10 100 10 =10 1000 100 =10 The ratios are the same, so the sequence is geometric. The initial term 𝑎 1 of the sequence is 0.001. As already observed, the common ratio r is 10. So the recursive rule is: 𝑎 1 =0.001 and 𝑎 𝑛 = 𝑎 𝑛−1 ∙10 The explicit rule is: 𝑎 𝑛 =0.001∙ (10) 𝑛−1

Ex: Given either an explicit or recursive rule for a geometric sequence, use a table to generate values and draw the graph of the sequence. 𝑎 𝑛 =2∙ 2 𝑛 for 𝑛≥0

Ex: Given either an explicit or recursive rule for a geometric sequence, use a table to generate values and draw the graph of the sequence. 𝑎 1 =10 and 𝑎 𝑛 =0.5∙ 𝑎 𝑛−1 for 𝑛≥2

Ex: Write a recursive rule and an explicit rule for the geometric sequence that models the situation. Then use the rule to answer the question. The Wimbledon Ladies’ Singles Championship begins with 128 players. Each match, two players play and only one moves to the next round. The players compete until there is one winner. How many rounds must the winner play? Identify the important information: The first round requires 64 matches, so a = 64. The next round requires half as many matches, so r = 1/2. Let n represent the number of rounds played and let 𝑎 𝑛 represent the number of matches played at that round. Create the explicit rule and the recursive rule for the tournament. The final round will have 1 match, so substitute this value into the explicit rule and solve for n. The explicit rule is: 𝑎 𝑛 =64∙ 1 2 𝑛−1 for 𝑛≥1. The recursive rule is: 𝑎 1 =64 and 𝑎 𝑛 = 1 2 ∙ 𝑎 𝑛−1 for 𝑛≥2. The final round will have 1 match, so substitute 1 for 𝑎 𝑛 into the explicit rule and solve for n. 𝑎 𝑛 =64∙ 1 2 𝑛−1 1=64∙ 1 2 𝑛−1 1 64 = 1 2 𝑛−1 1 2 6 = 1 2 𝑛−1 6=𝑛−1 𝑛=7 The winner must play in 7 rounds.