Arithmetic Sequences Objectives To identify and extend patterns

Arithmetic Sequences Objectives: To identify and extend patterns in sequences. To represent arithmetic sequences using functions notation

* A wooden post-and-rail fence with two rails is made as shown. Find the number of pieces of wood needed to build a 4 -section fence, a 5 section fence, and a 6 section fence. Suppose you want to build a fence with 3 rails. How many pieces of wood are needed for each size fence? Describe the pattern.

Sequence: an ordered list of numbers that often form a pattern. Term of a Sequence: Each number in the list

Problem #1: Extending Sequences Describe a pattern in each sequence. What are the next two terms of each sequence? A) B)

Problem #1 Got It? Describe a pattern in each sequence. What are the next two terms of each sequence. 1) 5, 11, 17, 23, … 2) 400, 200, 100, 50, … 3) 2, -4, 8, -16, … 4) -15, -11, -7, -3, …

Arithmetic Sequence: A number sequence formed by adding a fixed number to each previous term to find the next. Common Difference: The fixed number added to each previous term in an Arithmetic Sequence

Problem #2: Identifying an Arithmetic Sequence Tell whether the sequence is arithmetic. If it is, what is the common difference? A) 3, 8, 13, 18, … B) -3, -7, -10, -14, …

Problem #2 Got It? Tell whether each sequence is arithmetic. If it is, what is the common difference? 1) 8, 15, 22, 30, … 2) 7, 9, 11, 13, … 3) 10, 4, -2, -8, … 4) 2, -2, …

Recursive Formula: A function rule that relates each term of a sequence after the first to the ones before it Consider the sequence 7, 11, 15, 19…

Consider the sequence 7, 11, 15, 19… Common Difference:

Problem #3: Writing a Recursive Formula A) Write a recursive formula for the arithmetic sequence below. What is the value of the 8 th term? 70, 77, 84, 91, …

Problem #3: Writing a Recursive Formula B) Write a recursive formula for the arithmetic sequence below. What is the value of the 7 th term? 3, 9, 15, 21…

Problem #3 Got It? Write a recursive formula for each arithmetic sequence. What is the 9 th term of each sequence? 1) 23, 35, 47, 59, … 2) 7. 3, 7. 8, 8. 3, 8. 8, … 3) 97, 88, 79, 70, …

*Homework Textbook Page 279; #10 – 34 Even

Continued… Objectives: To represent arithmetic sequences using functions notation

Explicit Formula: A function rule that relates each term of a sequence to the term number The nth term of an arithmetic sequence with first term A(1) and common difference d is given by:

Problem #4: Writing an Explicit Formula A) An online auction works as shown below. Write an explicit formula to represent the bids as an arithmetic sequence. What is the 12 th bid?

Problem #4: Writing an Explicit Formula B) A subway pass has a starting value of $100. After one ride, the value of the pass is $98. 25. After two rides, its value is $96. 50. After three rides, its value is $94. 75. Write an explicit formula to represent the remaining value on the card as an arithmetic sequence. What is the value of the pass after 15 rides?

Problem #4: Writing an Explicit Formula C) Using your answer from B, how many rides can be taken with the $100 pass?

Problem #4 Got It? Justine’s grandfather puts $100 in a savings account for her on her first birthday. He puts $125, $150, and $175 into the account on her next 3 birthdays. If this pattern continues, how much will Justine’s grandfather put in the savings account on her 12 th birthday?

Problem #5: Writing an Explicit Formula From a Recursive Formula A) An arithmetic sequence is represented by the recursive formula A(n)=A(n – 1) + 12. If the first term of the sequence is 19, write the explicit formula.

Problem #5: Writing an Explicit Formula From a Recursive Formula B) An arithmetic sequence is represented by the recursive formula A(n)=A(n – 1) + 2. If the first term of the sequence is 21, write the explicit formula.

Problem #5 Got It? An arithmetic sequence is represented by the recursive formula A(n)=A(n – 1) + 7. If the first term of the sequence is 2, write the explicit formula.

Problem #6: Writing a Recursive Formula From an Explicit Formula A) An arithmetic sequence is represented by the explicit formula A(n)= 32 + (n – 1)(22). What is the recursive formula?

Problem #6: Writing a Recursive Formula From an Explicit Formula B) An arithmetic sequence is represented by the explicit formula A(n)= 76 + (n – 1)(10). What is the recursive formula?

Problem #6 Got It? An arithmetic sequence is represented by the explicit formula A(n)=1 +(n – 1)(3). Write the recursive formula.

*Homework Textbook Page 278 – 280; #1 – 8, 36 – 52 Even