Sequences & Series

Recursion Formula & Explicit FormulaChoiWhat 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

Example 1 Recursion FormulaGiven the recursion formulaa) Write the first 4 terms of each sequence.b)Identify the type of sequence.c) Find t20d) Find tn

The first 4 terms are {11, 7, 3,-1}Remark: To calculate any term we need to calculate all terms before it. This is a bit of a drawback.

22

33

44

b) Arithmetic Sequencea = 11 d = -4

c & d) RecallExplicit formula

2020

Example 2 Recursion FormulaGiven the recursion formulaa) Write the first 4 terms of each sequence.b)Identify the type of sequence.c) Find t15d) Find tn

The first 4 terms are {2, -6, 18,-54}

22

33

44

b) Geometric Sequencea = 2 r = -3

c & d) RecallExplicit formula

1515

Example 3 Construct Recursion FormulasConstruct a recursion formula to represent the following sequences: a) {5,7,9,11,13...}b) {-2, 6, -18, 54, ...}

= 5 +2Therefore, by observations, the recursion formula is:a) {5,7,9,11,13...}

= 7 +2

= 9 +2

= -2(-3) Therefore, by observations, the recursion formula is:b) {-2, 6, -18, 54,...}

=6(-3)

= -18(-3)

Geometric SequenceArithmetic SequenceExpect Common Difference (d)Expect Common Ratio (r)

Homework:Check the web siteHandout: Applications of SequencesMore Sequences Practice