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?
9-3
Sequences and Series
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Unit Objectives:
Write and use recursive definitions for a sequence.
Write and use explicit formulas for a sequence.
Determine if sequences are arithmetic or geometric.
Find values for sequences and series.
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 Series
Analyzing Arithmetic Sequences
Find the 46th term: Explicit Formula:
Find the 2nd and 3rd term of:
3, 5, 7, …
𝑎𝑛 = 𝑎 + (𝑛 − 1) ⋅ 𝑑 100, 94,
▒ , 88,
▒, 82, …
𝑎𝑛 = 3 + 𝑛 − 1 ⋅ 2
82 = 100 + 4 −1 ⋅ 𝑑
𝑎46 = 3 + 46 − 1 ⋅ 2 = 93
82 = 100 + 3𝑑
Find the 24th term:
4, 7, 10, …
𝑎24 = 4 + 24 − 1 ⋅ 3 = 73
−18 = 3𝑑
−6 = 𝑑
Finding missing term:
…, 15, 37,
▒ , 59, …
𝑥+𝑦
Arithmetic Mean:
2
15 + 59
2
Pg. 575
#9-60 by 3s
The number of seats in the first 13
rows in a section of this arena form
an arithmetic sequence.
How many seats are in row 13?
𝑎𝑛 = 𝑎(𝑛 − 1) ⋅ 𝑑
𝑎13 = 14 + 13 − 1 ⋅ 2 = 38