Download Arithmetic Sequences

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.
𝑥 2 + 7𝑥 + 12
𝑥+3
°
𝑥2
𝑥+4
=
𝑥 2 , 𝑥 ≠ −3, −4
Sequences and Series
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•
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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: 𝑎𝑛
n represents term number
Sequence:
Ordered list of numbers
1st Term 2nd Term 3rd Term
… n – 1 term nth term n + 1 term …
↓
𝑎1 ,
↓
𝑎2 ,
↓
𝑎3 ,
↓
…
𝑎𝑛−1 ,
2,
4,
6,
8,
…
↓
𝑎𝑛 ,
↓
𝑎𝑛+1 ,
…
Recursive Definition:
Explicit Formula: 𝑎𝑛 = 2𝑛 Uses the previous term (𝑎𝑛−1 ) 𝑎 =
2
1
Two
Parts:
Initial
Value
Describes sequence
Recursive Rule 𝑎𝑛 = 𝑎𝑛−1 +2
using term number (n)
Arithmetic Sequence
a, a + d, a + 2d, a + 3d, …
a = starting value
d = common difference
Recursive Definition:
𝑎1 = 𝑎
𝑎𝑛 = 𝑎𝑛−1 + 𝑑 for 𝑛 > 1
Explicit Formula:
𝑎𝑛 = 𝑎 + 𝑛 − 1 ⋅ 𝑑 for 𝑛 ≥ 1
4, 7, 10, 13, 16, …
+3
+3
+3
+3
Recursive Definition:
𝑎1 = 4
𝑎𝑛 = 𝑎𝑛−1 +3
Explicit Formula:
𝑎𝑛 = 4 + 𝑛 − 1 ⋅ 3
1, 4, 9, 16, 25, …
3 5 7 9
Not an Arithmetic Sequence
Analyzing Arithmetic Sequences
Find the 2nd and 3rd term of:
Find the 46th term: Explicit Formula:
▒ , 88,
▒, 82, …
3, 5, 7, …
𝑎𝑛 = 𝑎 + (𝑛 − 1) ⋅ 𝑑 100, 94,
𝑎𝑛 = 3 + 𝑛 − 1 ⋅ 2
𝑎46 = 3 + 46 − 1 ⋅ 2 = 93
Find the 24th term:
4, 7, 10, …
𝑎24 = 4 + 24 − 1 ⋅ 3 = 73
82 = 100 + 4 −1 ⋅ 𝑑
82 = 100 + 3𝑑
−18 = 3𝑑
−6 = 𝑑
Finding missing term:
…, 15, 37,
▒ , 59, …
Arithmetic Mean:
𝑎+𝑐
…, a, b, c, … b =
2
15 + 59
2
9-3
Geometric Sequences
Today’s Objective:
I can define, identify and apply geometric
sequences.
Geometric Sequence
a, a∙r, a∙r2, a∙r3, …
a = starting value
r = common ratio:
𝐶𝑢𝑟𝑟𝑒𝑛𝑡 𝑇𝑒𝑟𝑚
𝑃𝑟𝑒𝑣𝑖𝑜𝑢𝑠 𝑇𝑒𝑟𝑚
Recursive Definition:
𝑎1 = 𝑎
𝑎𝑛 = 𝑎𝑛−1 ⋅ 𝑟, for n > 1
Explicit Formula:
𝑎𝑛 = 𝑎 ⋅ 𝑟 𝑛−1 , for n ≥ 1
3, 6, 12, 24, 48, …
6
3
12 24 48
=2
6 12 24
Recursive
Definition:
𝑎1 = 3
𝑎𝑛 = 𝑎𝑛−1 ⋅ 2
Explicit Formula:
𝑎𝑛 = 3 ⋅ 2𝑛−1
𝑎1 = 2
𝑎𝑛 = 𝑎𝑛−1 ⋅ 4
2, 8, 32, 128, …
𝑎𝑛 = 2 ⋅ 4𝑛−1
Analyzing Geometric Sequences
Find the 10th term:
4, 12, 36, …
Find the 2nd and 3rd term of:
2, –▒6,, 18,
▒ , − 54, …
Explicit Formula:
𝑎𝑛 = 𝑎 ⋅ 𝑟 𝑛−1
Explicit Formula:
𝑎𝑛 = 𝑎 ⋅ 𝑟 𝑛−1
𝑛−1
𝑎𝑛 = 4 ⋅ 3
𝑎10 = 4 ⋅ 310−1
𝑎10 = 78,732
4 −1
−54 = 2 ⋅ 𝑟
−54 = 2 ⋅ 𝑟 3
−27 = 𝑟 3
−3 = 𝑟
Geometric Mean:
…, a, b, c, . . .
𝑏 2 = 𝑎𝑐
𝑏 = ± 𝑎𝑐
Finding the possible
missing term:
…, 48, ±12,
▒ , 3, …
𝑏 = ± 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
...
# of Red Triangles
1
3
9
27
...
Recursive Definition:
𝑎1 = 1
𝑎𝑛 = 𝑎𝑛−1 ⋅ 3
20
1,162,261,467
Explicit Formula:
𝑛−1
𝑎𝑛 = 1 ⋅ 3