Chapter 8 Sequences and Series
8.1 Arithmetic Sequences
Examples of Sequences:
Naming Sequences and Terms:
Sequences are Functions:
Recursive Formulas for Sequences:
Arithmetic (Linear) Sequences:
Ex 1: Identify 𝑎1 , the common difference 𝑑 and an explicit formula for each sequence
a. 1, 4, 7, 10, 13, 16, …
b. 15, 10, 5, 0, −5, −10, …
1
1
c. 1, 2 , 0, − 2 , −1, …
Ex 2: Find the common difference whose for the arithmetic sequence whose formula is 𝑎𝑛 = 6𝑛 + 3
Ex 3: Find the 81st term of the sequence 3, 5, 7, 9, …
Ex 4: Suppose 𝑎 is an arithmetic sequence with 𝑎2 = 14 and 𝑎6 = 4. Find
a. An explicit formula for 𝑎
b. A recursive formula for 𝑎
Ex 5: Suppose 𝑎 is an arithmetic sequence with 𝑎3 = −9 and 𝑎10 = −37
c. An explicit formula for 𝑎
d. A recursive formula for 𝑎
8.2 Geometric and Other Sequences
Geometric Sequences:
Ex 1: Write the first 5 terms of the geometric sequence in which 𝑎1 = 4 and 𝑟 = 5
Ex 2: Write the first 5 terms of the geometric sequence in which 𝑎𝑛 = 4𝑎𝑛−1 and 𝑎1 = 3.
Ex 3: Use the formula for the nth term of a geometric sequence to find the 6th term of the sequence with 𝑎1 =
1
15625 and 𝑟 = 5
Ex 4: If you invest $10,000 in a 5-year CD with an annual yield of 4.83%, the amount you begin with and the
amounts you have at the end of consecutive years form a finite geometric sequence.
a. Find an explicit formula for this sequence
b. Find a recursive formula for this sequence
Ex 5: For each sequence below
a. Determine whether it could be arithmetic, geometric, or neither
b. Give the next two terms of the sequence
c. State an explicit formula for the sequence
1. 𝑎𝑛 = 18, 24, 30, 36, …
2. 𝑏𝑛 = 16, 12, 9, 6.75
3. 𝑐𝑛 = 1, 3, 6, 10
Other Sequences:
1
4
9
16
Ex 6: Find an explicit formula for the nth term of a sequence beginning with 5 , 25 , 125 , 625 , …
Ex 7: Find an explicit formula for the nth term of a sequence beginning with 1, 8, 27, 64, 125, …
Arithmetic and Geometric Means
Ex 8: Find the missing term in each arithmetic sequence
a. … , 13, ____ , 7, …
4
68
b. … , − 7 , ____, − 21 , …
Ex 9: Find the missing term in each geometric sequence
a. … , −3, _____ , −108, …
b. … , −1, ____, −16, …
8.3 End Behavior of Sequences
End Behavior
Limit as n approaches infinity
Diverge vs. Converge
Ex 1: Find the limit of the following sequences. Then state whether it is convergent or divergent.
a. 𝑏𝑛 = 4𝑛 + 5
1
b. 𝑐𝑛 = 𝑛+3
c. 𝑑𝑛 = (−1)𝑛
d. 𝑒𝑛 =
1+4𝑛
𝑛
e. 𝑓𝑛 = 6
Properties of Limits and Constants
Ex 2: Find the limit of the sequence p where 𝑝𝑛 =
5𝑛+1
𝑛
Properties of Limits on Operations with Sequences
4𝑛+3
Ex 3: Consider 𝑝𝑛 = 𝑛 and 𝑞𝑛 =
a. Find lim 𝑝𝑛 and lim 𝑞𝑛
𝑛→∞
3𝑛−2
𝑛
𝑛→∞
b. Use the properties of limits to find lim (𝑝𝑛 ∙ 𝑞𝑛 )
𝑛→∞
8.4 Arithmetic Series
The Sum of Terms of a Sequence
Arithmetic Series
Sum of an Arithmetic Series
Ex 1: Find the sum of the arithmetic series 10 + 15 + 20 + 25 + 30 + 35 + 40 + 45 + 50
Ex 2: Find the sum of the arithmetic series 7 + 10 + 13 + … + 55
Ex 3: Find the sum of the arithmetic series 1 + 3·5 + 6 + 8·5 + . . . + 101
Ex 4: A student borrowed $8000 for college expenses. The load was to be repaid over a 100-month period,
with monthly payments as follows: $120.00, $119.50, $119.00, … , $70.50
a. How much did the student pay over the life of the loan?
b. What was the total interest paid on the loan?
Ex 5: The main floor of a new auditorium is planned to seat 800 people, with seats arranged in 20 rows. Each
row will have 2 more seats than the previous row. How many seat should there be in the first row?
8.5/8.7 Geometric Series and Infinite Series
Formula for the Sum of a Finite Geometric Series
Ex 1: A geometric series is given by 4 + (4 ∙ 3) + (4 ∙ 32 ) + (4 ∙ 33 ) + (4 ∙ 34 ). Evaluate 𝑆5 .
Ex 2: A geometric series is given by 1 − 4 + 16 − 64. Evaluate 𝑆9 .
1
Ex 3: Evaluate ∑6𝑛=1 24(3)𝑛−1
Ex 4: Evaluate
Infinite Series
4 8 16
2𝑛+1
4
Ex 5: Consider the infinite sequence 3 , 3 , 3 , … , 3 , … and its associated series 3 +
a. What type of sequence is this? Justify your answer.
8
+
3
16
3
+ …+
2𝑛+1
3
+. ..
b. Find the first five partial sums of the series.
c. Does the infinite series seem to converge?
Formula for the Sum of a Convergent Infinite Geometric Series
Ex 6: Which of the following could be a convergent geometric series? For those that could be, give the sum.
5 5 5 5
a. 3 , 9 , 27 , 81 , …
b.
5
3
+
5
9
5
+
27
+
5
81
+⋯
c. 640 + 160 + 40 + 10 + ⋯
d.
1
5
9
+2+2+
2
13
2