* Find the common difference for an arithmetic sequence. * Write terms of an arithmetic sequence. * Use a recursive formula for an arithmetic sequence. * Use an explicit formula for an arithmetic sequence.

1. 9.2 Arithmetic Sequences
Chapter 9 Sequences, Probability, and Counting
Theory

2. Concepts and Objectives
⚫ The objectives for this section are
⚫ Find the common difference for an arithmetic
sequence.
⚫ Write terms of an arithmetic sequence.
⚫ Use a recursive formula for an arithmetic sequence.
⚫ Use an explicit formula for an arithmetic sequence.

3. Arithmetic Sequences
⚫ An arithmetic sequence is a sequence in which one term
equals a constant added to the preceding term.
⚫ The constant for an arithmetic sequence is called the
common difference, d, because the difference between
any two adjacent terms equals this constant.
+
= −
1
n n
d a a

4. Arithmetic Sequences (cont.)
⚫ Formulas for calculating an for arithmetic sequences can
be found by linking the term number to the term value.
Example: The arithmetic sequence 3, 10, 17, 24, 31, …,
has a first term a1 = 3, and common difference d = 7:

5. Arithmetic Sequences (cont.)
⚫ Formulas for calculating an for arithmetic sequences can
be found by linking the term number to the term value.
Example: The arithmetic sequence 3, 10, 17, 24, 31, …,
has a first term a1 = 3, and common difference d = 7:
1 3
a =
2 3 7
a = +
( )( )
3 3 7 7 3 2 7
a = + + = +
( )( )
4 3 7 7 7 3 3 7
a = + + + = +
( )( )
3 1 7
n
a n
= + −

6. Arithmetic Sequences (cont.)
⚫ The nth term of an arithmetic sequence equals the first
term plus (n – 1) common differences. That is,
⚫ We can see that this is a linear function (where n is the
independent variable), and if we compare this to the
slope-intercept form, we can see that the slope is d, and
the y-intercept would be a0, or a1 – d if zero were in the
domain of the function (remember that our domain
is ).
( )
1 1
n
a a n d
= + −

7. Sequences - Examples
1. Calculate a100 for the arithmetic sequence
17, 22, 27, 32, …

8. Sequences - Examples
1. Calculate a100 for the arithmetic sequence
17, 22, 27, 32, …
d = 5
Therefore, ( )( )
100 17 100 1 5
a = + −
17 495 512
= + =

9. Sequences - Examples
2. The number 68 is a term in the arithmetic sequence
with a1 = 5 and d = 3. Which term is it?

10. Sequences - Examples
2. The number 68 is a term in the arithmetic sequence
with a1 = 5 and d = 3. Which term is it?
So, this is a22.
( )( )
68 5 1 3
n
= + −
( )( )
63 1 3
n
= −
21 1
n
= −
22
n =

11. Finding Common Differences
⚫ To determine whether a sequence is arithmetic, subtract
each term from the subsequent term. If each difference
is the same, then the sequence is arithmetic.
⚫ Example: Is each sequence arithmetic? If so, find the
common difference.
a) {1, 2, 4, 8, 16, …}
b) {–3, 1, 5, 9, 13, …}

12. Finding Common Differences
⚫ To determine whether a sequence is arithmetic, subtract
each term from the subsequent term. If each difference
is the same, then the sequence is arithmetic.
⚫ Example: Is each sequence arithmetic? If so, find the
common difference.
a) {1, 2, 4, 8, 16, …}
2 – 1 = 1, 4 – 2 = 2 Not arithmetic
b) {–3, 1, 5, 9, 13, …}
1 – (–3) = 4, 5 – 1 = 4, etc. Arithmetic

13. Finding Common Differences
⚫ If you know two consecutive numbers in an arithmetic
sequence, you can calculate the common difference. But
what if the terms aren’t consecutive?
⚫ Example: Find the common difference and a1 if we are
given a3 = 18 and a14 = 95.

14. Finding Common Differences
⚫ If you know two consecutive numbers in an arithmetic
sequence, you can calculate the common difference. But
what if the terms aren’t consecutive?
⚫ Example: Find the common difference a1 if we are given
a3 = 18 and a14 = 95.
Between n = 3 and n = 14, the common difference d has
been added 14 – 3 = 11 times. Therefore,
14 3 95 18 77
7
14 3 11 11
a a
d
− −
= = = =
−

15. Finding Common Differences
⚫ Example: Find the common difference and a1 if we are
given a3 = 18 and a14 = 95.
Between n = 3 and n = 14, the common difference d has
been added 14 – 3 = 11 times. Therefore,
and
14 3 95 18 77
7
14 3 11 11
a a
d
− −
= = = =
−
( )( )
1
1
18 3 1 7
18 14 4
a
a
= + −
= − =

16. Writing an Explicit Formula
⚫ Given the first several terms of an arithmetic sequence,
this is how to write an explicit formula:
1. Find the common difference, d = a2 – a1.
2. Substitute the common difference and the first term
into and simplify.
⚫ Example: Write an explicit formula for the arithmetic
sequence {2, 12, 22, 32, 42, …}.
( )
1 1
n
a a n d
= + −

17. Writing an Explicit Formula
⚫ Example: Write an explicit formula for the arithmetic
sequence {2, 12, 22, 32, 42, …}.
1. 12 – 2 = 10 = d
2.
3. So,
( )( )
2 1 10
2 10 10
8 10
n
a n
n
n
= + −
= + −
= − +
8 10
n
a n
= − +

18. Classwork
⚫ College Algebra 2e
⚫ 9.2: 6-24 (even); 9.1: 22-32 (even); 10.4: 42-48
(even)
⚫ 9.2 Classwork Check
⚫ Quiz 9.1