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Precalculus 9.2 9/17/13
Notes on Arithmetic Sequences HW: Pg 674 #2 – 5, 25, 27, 29, 37, 39
I. Arithmetic Sequences
A. Definition: A sequence whose successive terms differ by the same nonzero number d, called the
common difference. (this would be the slope of a linear function)
B. Refer to the patterns worksheet and write a function to find the nth term of the sequence for #1 and 2. a. Create a table b. Find the common difference (slope) c. Write an equation in point-slope form and solve d. Write function using subscript notation iii. Problem #1: 𝒂 𝒏
= 𝟑𝒏 − 𝟐 Problem #2: 𝒂 𝒏
= −𝟔𝒏 + 𝟏𝟏
C. In other words, the formula to find the nth term of an arithmetic sequence is LINEAR.
II. Notation
A. Using point-slope form to write the formula for the nth term of the arithmetic sequence.
Let’s convert this into a point-slope form that uses subscript notation: i. 𝑦 − 𝑦
1
= 𝑚(𝑥 − 𝑥
1
) → 𝑎 𝑛
− 𝑎
1
= 𝑑(𝑛 − 1) ii. The slope is the common difference d and the ordered pair (𝑥
1
sequence.
, 𝑦
1
) is the first term of the iii. Now solve this equation for 𝑎 𝑛
to get 𝑎 𝑛
= 𝑎
1
+ 𝑑(𝑛 − 1)
B. You can also use the slope-intercept form to find the nth term of the arithmetic sequence. i. 𝑦 = 𝑚𝑥 + 𝑏 → 𝑎 𝑛
= 𝑑𝑛 + 𝑐 ii. The slope is the common difference d. Substitute a term into the equation to find c.
II. Examples:
A. Finding terms given
sequence. 𝒂
𝟏
and d: find a formula for 𝑎 𝑛
and the first five terms of each arithmetic
1. The first term is 7 and the common difference is -3 
2. 𝑎
1
= −12, 𝑑 = 5 
B. Finding terms of an arithmetic sequence
1. Find 𝑎
13
and 𝑎 𝑛
for the arithmetic sequence -3, 1, 5, 9, …
= 9 and 𝑎
3
2. Find 𝑎
18
and 𝑎 𝑛
for the arithmetic sequence having 𝑎
2
= 15
3. Find 𝑎
1
given that the arithmetic sequence has the terms 𝑎
8
= −16 and 𝑎
16
= −40
4. Find the ninth term of the arithmetic sequence whose first two terms are 2 and 9

Pg. 674
9.2 Arithmetic Sequences
HW: Pg 674 #2 – 5, 25, 27, 29, 37, 39
Determine if the following sequences are arithmetic. If it is, find the common difference:
2. 10, 8, 6, 4, 2,..... 4. ,
5
2
, 2,
1
2
, 1,.....
6. -12, -8, -4, 0, 4, ...
Arithmetic d = -2
8. ln 1, ln 2, ln 3, ln 4, ln 5,......
Arithmetic d = -½ Arithmetic d = 4
10. 1 2 , 2 2 , 3 2 , 4 2 , 5 2 , ......
1, 4, 9, 16, 25...
Not Arithmetic Not Arithmetic
Find a formula for a n
for the given arithmetic sequences.
25.
,
3
2
, -1, -
7
2
, ...
d = -(5/2)
27. a
1
= 5, a
4
= 15 d = (10/3)
29. a
3
= 94, a d = -3
6
= 85 a n
= (5/2)n + (13/2) a n
= (10/3)n + (5/3) a
3
= a
1
+ 2d
94 = a
1
+ 2(-3) a
1
= 100 a n
= -3n + 103
Write the first 5 terms of the arithmetic sequence.
37. a
1
= 2, a
12
= 46 d = 4
2, 6, 10, 14, 18,...
39. a d = 4 a a
8
8
1
= 26, a
= a
26 = a
1
1
= -2
12
+ 7d
= 42
+ 7(4)
-2, 2, 6, 10, 14, ...