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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