Sequences F.LE.1, 2, 5 F.BF.1, 2 A.SSE.1.a F.IF.1, 2, 3, 4 http://www.cde.ca.gov/be/st/ss/documents/ccssmathstandardaug2013.pdf

Arithmetic Sequences: Writing Formulas Let’s examine Problem 1 on page 236:

Arithmetic Sequences: Writing Formulas Now let’s complete the table:

Arithmetic Sequences: Writing Formulas Now answer the questions on page 237: Explain how you can calculate the tenth term based on the ninth term. Determine the 20th term. Explain your calculation. Is there a way to calculate the 20th term without first calculating the 19th term? If so, describe the strategy. Describe a strategy to calculate the 93rd term.

Arithmetic Sequences: Explicit Formula General Rule Example A lowercase letter is used to name a sequence. 𝑎 The first term, or initial term, is referred to as 𝑎 1 . 𝑎 1 =125 𝑎 2 =143 𝑎 3 =161 The remaining terms are named according to the term number. A general term of the sequence is referred to as 𝑎 𝑛 , also known as the nth term, where n represents the index. 𝑎 𝑛 The term previous to 𝑎 𝑛 is referred to as 𝑎 𝑛−1 . 𝑎 𝑛−1 The common difference is represented as 𝑑. 𝑑=18 The INDEX is the position of the term (its term number) in a sequence.

Arithmetic Sequences: Explicit Formula The explicit formula for determining the nth term of an arithmetic sequence is: 𝑎 𝑛 = 𝑎 1 +𝑑 𝑛−1 nth term common difference 1st term previous term number

Arithmetic Sequences: Explicit Formula Let’s use an explicit formula to determine the 93rd term in the following situation: 𝑎 1 =125 and 𝑑=18 𝑎 𝑛 = 𝑎 1 +𝑑 𝑛−1 𝑎 93 =125+18 93−1 𝑎 93 =125+18 92 𝑎 93 =125+1656 𝑎 93 =1781 The 93rd term in the sequence is 1781. This means that Rico will contribute a total of $1781 if the Centipedes hit 92 home runs.

Arithmetic Sequences: Explicit Formula Complete the problems on page 239: Use the explicit formula to determine the amount of money Rico will contribute if the Centipedes hit: 35 home runs 48 home runs 86 home runs 214 home runs

Arithmetic Sequences: Explicit Formula What if Rico decides to increase his initial contribution to $500 and the amount donated per home run hit to $75.00. What would change? 𝑎 1 =500 and 𝑑=75

Arithmetic Sequences: Explicit Formula Complete the problems on page 240: Use the explicit formula to determine the amount of money Rico will contribute if the Centipedes hit: 11 home runs 26 home runs 39 home runs 50 home runs What are the first 10 terms of the sequence?

Geometric Sequences: Writing Formulas Now let’s read the problem on the bottom of page 240. How do you know the sequence 1, 2, 4, 8, 16, … is geometric? What is the common ratio? Now complete the table on page 241:

Geometric Sequences: Writing Formulas Now answer the questions on page 241: Explain how you can calculate the tenth term based on the ninth term. Determine the 20th term. Explain your calculation. Is there a way to calculate the 20th term without first calculating the 19th term? If so, describe the strategy.

Geometric Sequences: Explicit Formula General Rule Example A lowercase letter is used to name a sequence. 𝑔 The first term, or initial term, is referred to as 𝑔 1 . 𝑔 1 =1 𝑔 2 =2 𝑔 3 =4 The remaining terms are named according to the term number. A general term of the sequence is referred to as 𝑔 𝑛 , also known as the nth term, where n represents the index. 𝑔 𝑛 The term previous to 𝑔 𝑛 is referred to as 𝑔 𝑛−1 . 𝑔 𝑛−1 The common ratio is represented as 𝑟. 𝑟=2

Geometric Sequences: Explicit Formula The explicit formula for determining the nth term of a geometric sequence is: 𝑔 𝑛 = 𝑔 1 ∙ 𝑟 𝑛−1 nth term previous term number 1st term common ratio

Geometric Sequences: Explicit Formula Let’s use an explicit formula to determine the 20th term in the following situation: 𝑔 1 =1 and 𝑟=2 𝑔 𝑛 = 𝑔 1 ∙ 𝑟 𝑛−1 𝑔 20 =1∙ 2 20−1 𝑔 20 =1∙ 2 19 𝑔 20 =1∙524,288 𝑔 20 =524,288 The 20th term in the sequence is 524,288. This means that after 19 cell divisions, there are a total of 524,288 cells.

Geometric Sequences: Explicit Formula Complete the problems on page 243: Use the explicit formula to determine the total number of cells after: 11 divisions 14 divisions 18 divisions 22 divisions

Geometric Sequences: Explicit Formula What if we start with 5 different cells whose growth pattern changed such that the mother cell divided into 3 daughter cells? What would change? 𝑔 1 =5 and 𝑟=3

Geometric Sequences: Explicit Formula Complete the problems on page 244: Use the explicit formula to the total number of cells after: 4 divisions 7 divisions 13 divisions 16 divisions What are the first 10 terms of the sequence?

Sequences: Recursive Formulas A RECURSIVE FORMULA expresses each new term of a sequence based on the preceding term in the sequence. The recursive formula for determining the nth term of an arithmetic sequence is: 𝑎 𝑛 = 𝑎 𝑛−1 +𝑑 nth term previous term common difference

Sequences: Recursive Formulas Consider the following sequence: −2, −9, −16, −23, … Use the recursive formula to determine the 5th term 𝑎 𝑛 = 𝑎 𝑛−1 +𝑑 𝑎 5 = 𝑎 5−1 +(−7) 𝑎 5 = 𝑎 4 −7 𝑎 5 =−23−7 𝑎 5 =−30 The 5th term in the sequence is −30.

Sequences: Recursive Formulas The recursive formula for determining the nth term of a geometric sequence is: 𝑔 𝑛 = 𝑔 𝑛−1 ∙𝑟 nth term previous term common ratio

Sequences: Recursive Formulas Consider the following sequence: 4, 12, 36, 108, … Use the recursive formula to determine the 5th term 𝑔 𝑛 = 𝑔 𝑛−1 ∙𝑟 𝑔 5 = 𝑔 5−1 ∙ (3) 𝑔 5 = 𝑔 4 ∙ (3) 𝑔 5 =108∙ (3) 𝑔 5 =324 The 5th term in the sequence is 324.

Using Recursive Formulas Complete Problems 1 & 2 on pages 246 & 247 for homework.