What will the center number in Figure 6? 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? 6th = 84 10th = 220

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, 9-3 Today’s Objective: I can define, identify and apply arithmetic sequences. I can define, identify and apply geometric sequences.

Sequences Term of a Sequence: Sequence: Each number: 𝑓(𝑛) n represents term number Ordered list of numbers 1st Term 2nd Term 3rd Term … n – 1 term nth term n + 1 term ↓ f(1), f(2), f(3), … f(n-1), f(n), f(n+1), 2, 4, 6, 8, … Recursive Definition: Uses the previous term 𝑓(𝑛−1) Two Parts: Initial Value Recursive Rule Explicit Formula: Describes sequence using term number (n) 𝑓(𝑛)= 2𝑛 𝑓(1)= 2 𝑓(𝑛)= 𝑓 𝑛−1 +2

Arithmetic Sequence 4, 7, 10, 13, 16, … a, a + d, a + 2d, a + 3d, … +3 4, 7, 10, 13, 16, … a, a + d, a + 2d, a + 3d, … +3 +3 +3 +3 a = starting value Recursive Definition: 𝑓(1)= 𝑓(𝑛)= d = common difference 4 𝑓(𝑛−1) +3 Recursive Definition: 𝑓 1 =𝑎 𝑓(𝑛)=𝑓(𝑛−1)+𝑑 for 𝑛>1 Explicit Formula: 𝑓(𝑛)= 4 + 𝑛−1 ⋅3 1, 4, 9, 16, 25, … Explicit Formula: 𝑓(𝑛)=𝑎+ 𝑛−1 ⋅𝑑 for 𝑛≥1 3 5 7 9 Not an Arithmetic Series

Analyzing Arithmetic Sequences Find the 46th term: 3, 5, 7, … Explicit Formula: 𝑓(𝑛)=𝑎+(𝑛−1)⋅𝑑 Find the 2nd and 3rd term of: 100, ▒ , ▒, 82, … 94, 88, 𝑓(𝑛)= + 𝑛−1 ⋅2 82 = + −1 100 3 4 ⋅𝑑 82=100+3𝑑 𝑓(46)= 3+ 46−1 ⋅2 =93 −18=3𝑑 −6=𝑑 Find the 24th term: 4, 7, 10, … Finding missing term: …, 15, ▒ , 59, … 37, 𝑓(24)= 4+ 24−1 ⋅3 =73 Arithmetic Mean: …, a, b, c, … b= 𝑎+𝑐 2 15+59 2

I can define, identify and apply 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, … 6 3 12 6 3, 6, 12, 24, 48, … a, a∙r, a∙r2, a∙r3, … 6 3 12 6 24 12 48 24 = 2 a = starting value r = common ratio: 𝐶𝑢𝑟𝑟𝑒𝑛𝑡 𝑇𝑒𝑟𝑚 𝑃𝑟𝑒𝑣𝑖𝑜𝑢𝑠 𝑇𝑒𝑟𝑚 Recursive Definition: 𝑓(1)= 𝑓(𝑛)= Explicit Formula: 𝑓(𝑛)= 3 ⋅ 2 𝑛−1 Recursive Definition: 𝑓 1 =𝑎 𝑓(𝑛)=𝑓(𝑛−1)⋅𝑟, for n >1 3 𝑓(1)=2 𝑓(𝑛)=𝑓(𝑛−1)⋅4 𝑓(𝑛−1) ⋅2 2, 8, 32, 128, … 𝑓(𝑛)=2⋅ 4 𝑛−1 Explicit Formula: 𝑓(𝑛)=𝑎⋅ 𝑟 𝑛−1 , for n ≥ 1 Additional series for the board: 2, 4, 8, 16, … Geometric a(n) = 2*2^(n-1) 1, 5, 9, 13, 17, … No 2^3, 2^7, 2^11, 2^15, Geometric: a(n) = 2^3 * 2^4^(n-1) or 2^3*2^(4n-4) or 8*16^(n-1)

Analyzing Geometric Sequences Geometric Mean: …, a, b, c, . . . 𝑏 2 =𝑎𝑐 𝑏=± 𝑎𝑐 Find the 10th term: 4, 12, 36, … Find the 2nd and 3rd term of: 2, ▒ , ▒ , − 54, … – 6, 18, Explicit Formula: Explicit Formula: 𝑓 𝑛 =𝑎⋅ 𝑟 𝑛−1 𝑓(𝑛)=𝑎⋅ 𝑟 𝑛−1 Finding the possible missing term: …, 48, ▒ , 3, … 𝑓(𝑛)= 4 ⋅ 3 𝑛−1 −54 =2 4 ⋅𝑟 −1 𝑓(10)= 4⋅ 3 10−1 −54=2⋅ 𝑟 3 ±12, −27= 𝑟 3 𝑓(10)= 78,732 −3=𝑟 𝑏=± 48⋅3 =± 144 =±12

Sierpinski Triangle p. 575: 7-23 odd, 41-49 odd Stage 1 Stage 2 Stage 3 Stage 4 How many red triangles are there at stage 20? Stage 1 2 3 4 . . . # of Red Triangles 20 1,162,261,467 1 3 9 27 Recursive Definition: 𝑓(1)= 𝑓 𝑛 = Explicit Formula: 𝑓 𝑛 = 1 1 ⋅ 3 𝑛−1 𝑓(𝑛−1) ⋅3