1. Grade 11 Functions (MCR3U)
Unit 5: Pascal’s Triangle, Binomial Theorem, Sequences & Series Recursion & Explicit Formulas
Mr. Choi
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2. What is a Recursion Formula?
A recursion formula consists of at least 2 parts.
One part gives the value(s) of the first term(s) in the sequence,
and the other part gives an equation that can be used to
calculate each of the other terms from the term(s) before it.
n must be integers, normally natural numbers
Recursion & Explicit Formulas
© 2018 E. Choi – MCR3U - All Rights Reserved

3. Example 1 – Recursion Formula
Given the recursion formula a) Write the first 4 terms of each sequence. b) Identify the type of sequence. c) Find t20 d) Find tn
b) Arithmetic Sequence
a = d = -4
Remark: To calculate any term we need to calculate all terms before it. This is a bit of a drawback.
c & d) Recall 4 4 2 2 3 3
Explicit formula 20 20
The first 4 terms are {11, 7, 3,-1}
Recursion & Explicit Formulas
© 2018 E. Choi – MCR3U - All Rights Reserved

4. Example 2 – Recursion Formula
Given the recursion formula a) Write the first 4 terms of each sequence. b) Identify the type of sequence. c) Find t15 d) Find tn
b) Geometric Sequence
a = r = -3
c & d) Recall
Explicit formula 4 4 2 2 3 3 15 15
The first 4 terms are {2, -6, 18,-54}
Recursion & Explicit Formulas
© 2018 E. Choi – MCR3U - All Rights Reserved

5. Example 3 – Construct Recursion Formulas
Construct a recursion formula to represent the following sequences: a) {5,7,9,11,13...} b) {-2, 6, -18, 54, ...}
Expect Common Ratio (r)
Expect Common Difference (d)
b) {-2, 6, -18, 54,...}
a) {5,7,9,11,13...}
Geometric Sequence
Arithmetic Sequence
= -2(-3)
= 5 +2
= 7 +2
=6(-3)
= -18(-3)
= 9 +2
Therefore, by observations, the recursion formula is:
Therefore, by observations, the recursion formula is:
Recursion & Explicit Formulas
© 2018 E. Choi – MCR3U - All Rights Reserved

6. Homework: Text Book: P. 370 #1acd,2ace,3-14,16,18 Work Sheet:
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Recursion & Explicit Formulas
© 2018 E. Choi – MCR3U - All Rights Reserved

7. End of Lesson Recursion & Explicit Formulas
© 2018 E. Choi – MCR3U - All Rights Reserved