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12.2 – Analyze Arithmetic Sequences and Series
Arithmetic Sequence:
The difference of consecutive terms is constant
Common Difference:
d, the difference between any two terms
1. Tell whether the sequence is arithmetic.
–4, 1, 6, 11, 16, …
+5
Yes
+5 +5 +5
a.
1. Tell whether the sequence is arithmetic.
3, 5, 9, 15, 23, …
+2 +4 +6 +8
No
b.
1. Tell whether the sequence is arithmetic.
17, 14, 11, 8, 5 ,…
–3
Yes
–3 –3 –3
c.
Rule for an Arithmetic Sequence
The nth term of an arithmetic sequence with first term a1 and common difference d is given by:
1 1na a n d
2.Write a rule for the nth term of the sequence. Then find a15.
4, 9, 14, 19, …+5
a.
1 1na a n d
d = 5
a1 = 4
4 1 5na n 4 5 5na n
5 1na n
15a 5 15 1
15 75 1a
15 74a
2.Write a rule for the nth term of the sequence. Then find a15.
60, 52, 44, 36, …–8
b.
1 1na a n d
d = –8
a1 = 60
60 1 8na n 60 8 8na n
8 68na n
15a
15 120 68a
15 52a
8 15 68
3. Write a rule for the nth term of the sequence given:
1 1na a n d
d = 3, a19 = 48
19 1 19 1 3a a 148 18 3a
148 54a
16 a
6 1 3na n
6 3 3na n
3 9na n
4. Write a rule for the nth term of the sequence given:
1 1na a n d
d = –7, a11 = –57
11 1 11 1 7a a 157 10 7a
157 70a
113 a
13 1 7na n
7 20na n
13 7 7na n
5. Write a rule for the nth term of the sequence given:
1 1na a n d a8 = 21, a27 = 97
121 8 1a d
197 27 1a d 121 7a d
197 26a d
–1
121 7a d
76 19d4 d
121 7 4a 17 a
7 1 4na n 7 4 4na n
4 11na n
6. Write a rule for the nth term of the sequence given:
1 1na a n d a7 = 26, a16 = 71
126 7 1a d
171 16 1a d 126 6a d
171 15a d
–1
126 6a d
45 9d5 d
126 6 5a 14 a
4 1 5na n 4 5 5na n
5 9na n
Arithmetic Series:
Adding the terms of an arithmetic sequence
Sum of a Finite Arithmetic Series:
1
2n
na a
S n
The sum of the first n terms is:
1
2n
na a
S n
7. What is the sum of the arithmetic series?
20
1
3 5i
i
8 10320
2S
a1 = 3+5(1)
a1 = 8a20 =3+5(20)
a20 =103
20 55.5S
1110S
n = 20
1
2n
na a
S n
8. What is the sum of the arithmetic series?
9
4
6 30i
i
6 246
2S
a4 = 6(4) – 30
a4 = –6a9 = 6(9) – 30
a20 = 24
6 9S
54S
n = 6
1
2n
na a
S n
9. What is the sum of the arithmetic series?
16
8
11 4i
i
21 539
2S
a8 = – 11 + 4(8)
a8 = 21a16 =– 11 + 4(16)
a16 = 53
9 37S
333S
n = 9
1
2n
na a
S n
10. What is the sum of the arithmetic series?5 7 9 11 ... 203
5 203100
2S
a1 = 5
an = 203
100 104S
10,400S
1 1na a n d
d = 2
203 2 1 2n 203 5 2 2n
203 2 3n
100 n200 2n
1
2n
na a
S n
11. What is the sum of the arithmetic series?2 9 16 23 ... 198
2 19829
2S
a1 = 2
an = 198
29 100S
2,900S
1 1na a n d
198 2 1 7n 198 2 7 7n
198 7 5n
29 n
d = 7
203 7n
