Chapter 9 1 Mathematical Patterns St Augustine Preparatory

Chapter 9 -1 Mathematical Patterns St. Augustine Preparatory School March 1, 2016

Note: • These notes do not include information on finding recursive formulas for non-arithmetic and non-geometric sequences (as discussed in class).

Definitions • Sequence: An ordered set of numbers • Term: Each number in a sequence • Arithmetic sequence: is a sequence with the difference between two consecutive terms constant. The difference is called the common difference. • Geometric sequence: is a sequence with the ratio between two consecutive terms constant. This ratio is called the common ratio.

What type of sequence? 1) 2) 3) 4) 5) 3, 8, 13, 18, 23 1, 2, 4, 8, 16 24, 12, 5, 3, 3/2, 3/4 55, 51, 47, 43, 39, 35 1, 5, 14, 30, 55

What type of sequence? 1) 2) 3) 4) 5) 3, 8, 13, 18, 23 A. S. ; Common Difference: 5 1, 2, 4, 8, 16 G. S. ; Common ratio: 2 24, 12, 5, 3, 3/2, 3/4 G. S. ; CR: 1/2 55, 51, 47, 43, 39, 35 A. S. ; CD: -4 1, 5, 14, 30, 55 Neither

Recursive Formulas • A recursive formula requires the computation of previous terms in order to find term an • Example of a recursive formula: a 1 = 133 and an = an-1 – 3 For a recursive formula you need: 1. To state the first term (a 1 = 133 for the example above). 2. A recursive formula (an = an-1 – 3)

Generating a Sequence from a Recursive Formula Example: an = an-1 – 3, where a 1 = 133 a 2 = a 2 -1 – 3 = a 1 – 3 = 133 – 3 = 130 a 3 = a 3 -1 – 3 = a 2 – 3 = 130 – 3 = 127 You can see we need the term an-1 in order to calculate an.

Writing a Recursive Formula for an Arithmetic Sequence • Example: 1, 3, 5, 7, 9 , 11 a 2 – a 1 = 3 – 1 = 2 a 3 – a 2 = 5 – 3 = 2 a 4 - a 3 = 7 – 5 = 2 a 5 – a 4 = 9 – 7 = 2 We need both a recursive formula and the first term: a 1 = 1 and an = an-1 + 2

Writing recursive formulas for geometric sequences 3, 6, 12, 24, 48, 96 a 1 = 3 a 2/a 1 = 6/3 = 2 a 3/a 2 = 12/6 = 2 a 4/a 3 = 24/12 = 2 (at this point, you can see. . ) a 1 = 3 and an = 2 an-1