Chapter 1: Number Patterns 1.2: Mathematical Patterns Essential Question: How can recognizing number patterns help in making a decision?

1.2 Mathematical Patterns Sequence → ordered list of number Each number in the list is called a “term” First term of a sequence is denoted u 1 Second term: u 2 The n th term in the sequence: u n The term before u n is u n-1

1.2 Mathematical Patterns Example 1: Terms of a Sequence. Solution: We’re adding two circles to the previous diagram, one at the top, one at the bottom: Diagram 1 1 circle Diagram 2 3 circles Diagram 3 5 circles ○○○ ○ ○○○ ○ Diagram 4 7 circles Diagram 5 9 circles ○○○○ ○ ○○○○○ ○

1.2 Mathematical Patterns The number of circles in the diagrams can be represented by the sequence: {1, 3, 5, 7, 9, … } which can be expressed using sequence notation u 1 = 1 u 2 = 3 u 3 = 5 u 4 = 7 u 5 = 9 … u n = u n ▫A number in the sequence is equal to the previous number + 2 A sequence is a function, because each input corresponds to exactly one output. ▫The domain of a sequence is a subset of the integers ▫The range is the set of terms of the sequence The above sequence has a domain of { 1, 2, 3, 4, 5, … } and has a range of { 1, 3, 5, 7, 9, … }. A graph of the function is on page 15.

1.2 Mathematical Patterns Recursive Form of a Sequence ▫A sequence is defined recursively if:  the first term is given, and  there is a method of determining the n th term by using the terms that precede it. ▫Give the starting term and the pattern to determine the rest of the sequence ▫Example 3: Recursively Defined Sequence  Define the sequence { -7, -4, -1, 2, 5, … } recursively Answer: u 1 = -7 and u n = u n for n ≥ 2

1.2 Mathematical Patterns Alternate Sequence Notation ▫Sometimes it’s more convenient to begin numbering a sequence with a number other than 1 (usually 0) ▫Example 4: Using Alternate Sequence Notation  Ball starts at 9 feet, and bounces to 2/3 its height each rebound

1.2 Mathematical Patterns Example 5: Salary Raise Sequence ▫Starting salary of $20,000. ▫A raise of $2000 earned at the end of each year ▫Find the value at the end of the 6 th year ▫Show using the calculator Recursive Sequence: u 0 = $20,000 and u n = u n u 0 = $20,000 u 1 = u 0 + $2000 = $20,000 + $2000 = $22,000 u 2 = u 1 + $2000 = $22,000 + $2000 = $24,000 u 3 = u 2 + $2000 = $24,000 + $2000 = $26,000 u 4 = u 3 + $2000 = $26,000 + $2000 = $28,000 u 5 = u 4 + $2000 = $28,000 + $2000 = $30,000 u 6 = u 5 + $2000 = $30,000 + $2000 = $32,000

1.2 Mathematical Patterns Example 6: Adding a Pattern of Values ▫1 chord in a circle → 2 regions ▫2 chords → 4 regions ▫3 chords → 7 regions ▫4 chords → 11 regions Find a recursive function to represent the maximum number of regions formed with n chords Answer: u 1 = 2 and u n = u n-1 + n for n ≥ 2

1.2 Mathematical Patterns Example 7: Adding Chlorine to a Pool ▫Start with 3.4 gal of chlorine in the pool ▫Adds 0.25 gallons at beginning of each day ▫15% evaporates each day ▫How much chlorine will be in the pool on the 6 th day Approximately 2.3 gallons uu 1 = 3.4 and u n = 0.85(u n ) SShortcut: Replace u n-1 with ‘ANS’ on calculator & keep track of iterations

1.2: Mathematical Patterns Assignment ▫Page 19 ▫5-21 (all problems) ▫Ignore all directions to graph the sequence