Grade 11 Functions (MCR3U) Unit 5: Pascal’s Triangle, Binomial Theorem, Sequences & Series Recursion & Explicit Formulas Mr. Choi © 2018 E. Choi – MCR3U - All Rights Reserved

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

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 = 11 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

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 = 2 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

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

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

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