Number Patterns and Sequences
Explore the relationship between numbers in center tiles of figures, solve arithmetic sequence problems, and understand geometric sequences. Enhance your math skills with this comprehensive guide.
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Presentation Transcript
For each figure, how is the number on the center tile related to the numbers on the other tiles? What will the center number in Figure 6? What will the center number be in figure 10?
Warm Up Simplify and state restrictions. ?2 ?2+ 7? + 12 ? + 3 = ?2, ? 3, 4 ? + 4
Sequences and Series Unit Objectives: Describe a list of numbers using sequence/series terminology Write recursive definitions, explicit formulas and summation notation for sequences/series Find values for arithmetic/geometric sequences/series. Model problems using sequences/series 9-2 Today s Objective: I can define, identify and apply arithmetic sequences.
Sequences Term of a Sequence: Each number: ?? n represents term number n 1 term nth term n + 1 term ?? 1, ??, Sequence: Ordered list of numbers 1st Term 2nd Term ?1, 2, 3rd Term ?3, 6, ?2, 4, ??+1, 8, Recursive Definition: Uses the previous term (?? 1) Two Parts: Initial Value Recursive Rule Explicit Formula: Describes sequence using term number (n) ??= 2? ?1= 2 ??= ?? 1+2
Arithmetic Sequence 4, 7, 10, 13, 16, a, a + d, a + 2d, a + 3d, +3 +3 +3 +3 Recursive Definition: ?1= ??= ?? 1 a = starting value d = common difference 4 +3 Recursive Definition: ?1= ? ??= ?? 1+ ? for ? > 1 Explicit Formula: ??= 4 + ? 1 1, 4, 9, 16, 25, 3 Explicit Formula: ??= ? + ? 1 ? for ? 1 5 9 7 3 Not an Arithmetic Sequence
Analyzing Arithmetic Sequences Find the 46th term: 3, 5, 7, ??= ? + (? 1) ? 2 Find the 2nd and 3rd term of: 100, , , 82, 94, 88, Explicit Formula: + 1 4 ? = 100 ??= 82 3 + ? 1 82 = 100 + 3? ?46= 3 + 46 1 2 = 93 18 = 3? 6 = ? Finding missing term: , 15, , 59, 37, Find the 24th term: 4, 7, 10, = 73 ?24= 4 + 24 1 3 Arithmetic Mean: , a, b, c, b = 15 + 59 2 ?+? 2
9-3 Geometric Sequences Today s Objective: I can define, identify and apply geometric sequences.
Geometric Sequence 3, 6, 12, 24, 48, a, a r, a r2, a r3, 12 6 6 3 48 24 24 12 = 2 a = starting value r = common ratio: ??????? ???? Recursive Definition: ?1= ??= Explicit Formula: ??= 2? 1 3 ???????? ???? Recursive Definition: ?1= ? ??= ?? 1 ?, for n > 1 3 ?? 1 2 ?1= 2 ??= ?? 1 4 ??= 2 4? 1 2, 8, 32, 128, Explicit Formula: ??= ? ?? 1, for n 1
Analyzing Geometric Sequences Geometric Mean: , a, b, c, . . . ?2= ?? ? = ?? Find the 10th term: 4, 12, 36, Find the 2nd and 3rd term of: 2, , , 54, 6, 18, Explicit Formula: ??= ? ?? 1 4 Explicit Formula: ??= ? ?? 1 Finding the possible missing term: , 48, , 3, 12, 1 3? 1 ??= 4 = 2 54 ? 54 = 2 ?3 4 310 1 ?10= ?10= 27 = ?3 3 = ? 78,732 ? = 48 3 = 144 = 12
Sierpinski Triangle p. 575: 7-23 odd, 41-49 odd p. 584: 7-17 odd, 33-43 odd Stage 4 Stage 1 Stage 2 Stage 3 How many red triangles are there at stage 20? Stage 1 2 3 4 . . . 20 # of Red Triangles 1,162,261,467 . . . 1 3 9 27 Recursive Definition: ?1= ??= ?? 1 Explicit Formula: ??= 1 1 3? 1 3
