Objectives:

To identify and extend patterns in sequences.

To represent arithmetic sequences using functions notation

Section 4-7 Arithmetic Sequences

*Warm up: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.

Key VocabularySequence:

an ordered list of numbers that often form a pattern.

Term of a Sequence: Each number in the list

Problem #1: Extending SequencesDescribe a pattern in each sequence. What are the next two terms of each sequence?

A)

B)

Problem #1Got 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, …

More VocabularyArithmetic 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 #2Got 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, …

More VocabularyRecursive 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 FormulaA) 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 FormulaB) Write a recursive formula for the arithmetic sequence below. What is the value of the 7th term?

3, 9, 15, 21…

Problem #3Got 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, …

*HomeworkTextbook Page 279; #10 – 34 Even

Objectives:

To represent arithmetic sequences using functions notation

Section 4-7 Continued…

Key VocabularyExplicit Formula:

A function rule that relates each term of a sequence to the term numberThe nth term of an arithmetic sequence

with first term A(1) and common difference d is given by:

7, 11, 15, 19,…

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 #4Got 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

FormulaA) 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

FormulaB) 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 #5Got 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

FormulaA) 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

FormulaB) An arithmetic sequence is represented by the explicit formula A(n)= 76 + (n – 1)(10). What is the recursive formula?

Problem #6Got It?

An arithmetic sequence is represented by the explicit formula A(n)=1 +(n – 1)(3). Write the recursive formula.

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