1. Warm-Up Write the first five terms of an = 4n + 2 a1 = 4(1) + 2
= st Term
a2 = 4(2) + 2
= nd Term
a3 = 4(3) + 2
= rd Term
a4 = 4(4) + 2
= th Term
a5 = 4(5) + 2
= th Term

2. Analyze Arithmetic Sequences and Series
Section 12-2
Analyze Arithmetic Sequences and Series

3. Vocabulary
Arithmetic Sequence – The difference of consecutive terms is constant.
Common Difference – The constant difference, denoted by d.
Arithmetic Series – The expression formed by adding the terms of an arithmetic sequence.

4. Rule for an Arithmetic Sequence
The nth term of an arithmetic sequence with first term a1 and the common difference d is given by:
an = a1 + (n – 1)d
The Sum of a Finite Arithmetic Series The sum of the first n terms of an arithmetic series is:
a1 + an 2
Sn is the mean of the 1st and nth terms, multiplied by the number of terms.
Sn = n

5. Example 1 Tell whether the sequence is arithmetic.
b.) -8, -4, 0, 6, 12, …
a2 – a1 =
13 – 7 = 6
YES
a3 – a2 =
19 – 13 = 6
a4 – a3 =
25 – 19 = 6
a2 – a1 =
-4 – (-8) = 4 NO
a3 – a2 =
0 – (-4) = 4
a4 – a3 =
6 – 0 = 6

6. Example 2
Write a rule for the nth term of the sequence. Then find a20.
a.) -7, -10, -13, -16, …
an = a1 + (n – 1)d
a1 = -7
d = -3
an = -7 + (n – 1)(-3)
an = n + 3
Rule: an = n
a20 = (20)
a20 = -64

7. Example 2 - continued
Write a rule for the nth term of the sequence. Then find a20.
b.) 59, 68, 77, 86, …
an = a1 + (n – 1)d
a1 = 59
d = 9
an = 59 + (n – 1)(9)
an = n – 9
Rule: an = n
a20 = (20)
a20 = 230

8. Example 3
One term of an arithmetic sequence is a27 = The common difference is d = 11.
a.) Write the rule for the nth term.
an = a1 + (n – 1)d
a27 = a1 + (27 – 1)(11)
263 = a
a1 = -23
an = (n – 1)11
an = n – 11
Rule: an = n

9. Example 3 - continued b.) Graph the first 6 terms of the sequence.
Create a table of values for the sequence.
Rule: an = n an n 1 2 3 4 5 6
-23
a1 = -23
-12
a2 = (2) -1
a3 = (3) 10
a4 = (4) 21
a5 = (5) 32
a6 = (6)

10. Example 3 - continued b.) Graph the first 6 terms of the sequence. an30 25 an n 1 2 3 4 5 6 20
-23 15
-12 10 5 -1 -1 1 2 3 4 5 6 10 -5 21
-10 32
-15
-20
-25
-30

11. Example 4
Two terms of an arithmetic sequence are a10 = 148 and a44 = Find a rule for the nth term.
Step 1: Write a system of equations using
an = a1 + (n – 1)d and substitute 44 for n in equation 1 and then 10 for n in equation 2.
a44 = a1 + (44 – 1)d
556 = a1 + 43d
a10 = a1 + (10 – 1)d
148 = a1 + 9d
408 = 34d
Step 2: Solve the system.
d = 12
Step 3: Substitute d into equation 1.
556 = a1 + 43(12)
a1 = 40

12. Example 4 - continued
Two terms of an arithmetic sequence are a10 = 148 and a44 = Find a rule for the nth term.
Step 4: Find a rule for an an = a1 + (n – 1)d
a1 = 40
d = 12
an = 40 + (n – 1)(12)
an = n – 12
an = n

13. Example 5 ∑ Find the sum of the arithmetic series a1 = -2 + 4i28 ∑
(-2 + 4i)
i = 1
a1 = i
Step 1: Identify the first term.
a1 = (1)
= 2
Step 2: Identify the last term.
a28 = i
a28 = (28)
= 110
Step 3: Write the rule for S28 , substitute 2 for a1 and 110 for a28.
S28 = 2 28
Sn = a1 + an 2 n
S28 = 1568

14. Homework Section 12-2 Page 806 - 807 5 – 10, 15 – 18,
23 – 25, 29, 31 – 33,
39, 40 – 43, 46,
55 – 57