Common Core Algebra Unit 12: Sequences
3/3/17
Lesson 5: Recursive Formulas
Objective: SWBAT write a recursive formula and apply the formula to determine terms in a sequence.
Do Now: Create a sequence of numbers using the following information below. Write the sequence here:
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1) The first number of the sequence is 2.
2) To find the second number of the sequence, take the first number, multiply it by 5 then subtract 3.
3) To find the third number of the sequence, take the second number, multiply it by 5 then subtract 3.
4) To find the fourth number of the sequence, take the third number, multiply it by 5 then subtract 3.
Turn and Talk:
Let’s take a closer look at the sequence from the Do Now.
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1) Is the sequence arithmetic?
2) Is the sequence geometric?
3) Does the sequence follow a pattern?
The sequence from the Do Now can be defined _______________________.
Using a Recursive Rule to Generate a Sequence
A recursive rule for a sequence defines the nth term by relating it to one or more previous terms.
A recursive formula will help you find the next term in a sequence. Each term is found by doing something (+, , x, ÷) to the previous term(s).
A recursive formula is written with two parts:
 a statement of the starting term
 a statement of the formula used to arrive at the next term
Let’s define the sequence from the Do Now recursively.
The first term is 2.
The nth term equals 5 times the previous term minus 3.
Model Example:
Find the first four terms of each sequence given the
recursive rule.
a1 = 6; an = 2an-1 + 1
Check for Understanding:
Find the first four terms of each sequence given the
recursive rule.
a1 = 12; an = ½an-1 - 4
Thinking back to yesterday’s lesson…
Determine if the sequence below is arithmetic or geometric. For each sequence write an explicit rule that can be
used to find the nth term of the sequence.
a) 4, 7, 10, 13, ...
b) 1, 3, 9, 27, ...
Arithmetic and Geometric Sequences can be defined Recursively & Explicitly:
Let’s take a closer look at the sequences from the previous example .
Can the sequence 4, 7, 10, 13, ... be defined with a recursive rule?
an =
Can the sequence 1, 3, 9, 27, ... be defined with a recursive rule?
an =
Writing Rules to Generate Arithmetic and Geometric Sequences:
Model Example:
Write a recursive formula for the following sequence.
100, 96, 92, ...
Check for Understanding:
Write a recursive formula for the following sequence.
200, 40, 8, ...
Model Example:
Find the first 3 terms in each sequence below.
Check for Understanding:
Find the first 3 terms in each sequence below.
Lesson Summary:
Sequences defined recursively use the _____________________ term(s) to find the next term of the sequence.
When sequences are defined explicitly, any term of the sequence can be found using the explicit formula.
Problem Set:
1. Find the first four terms of each sequence given the recursive rule.
a) f(1) = -4; f(n) = f(n-1) – 6
b) f(1) = 1/2 ; f(n) = -4f(n-1) + 6
2. If a sequence is defined recursively by f(1) = 10 and f(n) = -f(n-1) + 3 then find f(5).
3. A sequence is defined recursively by adding 4 to twice the value of the previous term. Write a recursive rule
for this sequence if the first term is 15.
4. The figure below represents the first three terms of a sequence.
Which of the following rules can be used to define the sequence? Select all that apply.
Justify your response.
.Complete all written portions on the lines provided.
7) What is the difference between a recursive formula and an explicit formula?
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8) If you were asked to find the 67th term in a sequence, which formula (Explicit/Recursive) would best suit
the task, and why?
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Name _______________________
Common Core Algebra Unit 12: Sequences
Exit Slip
3/3/17
Lesson 5: Recursive Formulas
Objective: SWBAT write a recursive formula and apply the formula to determine terms in a sequence.
Given the first 3 terms, determine if the sequence is arithmetic or geometric. Then write the recursive formula.
2, 9, 16….
Name _______________________
Exit Slip
Common Core Algebra Unit 12: Sequences
3/3/17
Lesson 5: Recursive Formulas
Objective: SWBAT write a recursive formula and apply the formula to determine terms in a sequence.
Given the first 3 terms, determine if the sequence is arithmetic or geometric. Then write the recursive formula.
2, 9, 16….