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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 6section 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, 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 8th 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 7th term? 3, 9, 15, 21… Problem #3 Got It? Write a recursive formula for each arithmetic sequence. What is the 9th 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 12th 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 12th 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