1. Defining and Using Sequences and Series
Section 8.1 beginning on page 410

2. Sequences
A sequence is an ordered list of numbers. A finite sequence has an end and its domain is the set {1,2,3,…,𝑛}. The values in the range are called the terms of the sequence.
An infinite sequence is a function that continues without stopping and whose domain is the set of positive integers.
Finite sequence: 2,4,6,8 Infinite sequence: 2,4,6,8,…
A sequence can be specified by an equation, or a rule. For example, both sequences above can be described by the rule
𝑎 𝑛 =2𝑛 or
𝑓 𝑛 =2𝑛

3. Writing the Terms of a Sequence
Note: The domain of a sequence could begin with 0 instead of 1. Unless otherwise indicated, assume the domain of a sequence begins with 1.
Example 1: Write the first six terms of (a) and (b).
a) 𝑎 𝑛 =2𝑛+5 b) 𝑓 𝑛 = (−3) 𝑛−1
𝑎 1 =
2(1)+5 =7
First Term
𝑓(1)=
(−3) 1 −1 =1
(−3) 2 −1
=−3
𝑎 2 =
2(2)+5 =9
Second Term
𝑓(2)= =9
𝑎 3 =
2(3)+5
=11
Third Term
𝑓(3)=
(−3) 3 −1
𝑎 4 =
2(4)+5
=13
Fourth Term
𝑓(4)=
(−3) 4 −1
=−27
𝑎 5 =
2(5)+5
=15
Fifth Term
𝑓(5)=
(−3) 5 −1
=81
𝑎 6 =
2(6)+5
=17
Sixth Term
𝑓(6)=
(−3) 6 −1
=−243

4. Writing Rules for Sequences
Example 2a: Describe the pattern, write the next term, and write a rule for the nth term of the sequence.
𝑛=2
𝑛=3
𝑛=4
𝑛=1
−1,−8,−27,−64,…
How are these terms related to each other? Can they be defined in a similar way?
The terms are all perfect cubes.
The next term would be (−5) 3 =−125
To write a rule, relate the terms to their relative position (the n values).
(−1) 3 , (−2) 3 , −3 3 , (−4) 3
𝑎 𝑛 =
(−𝑛) 3

5. Writing Rules for Sequences
Example 2b: Describe the pattern, write the next term, and write a rule for the nth term of the sequence.
𝑛=2
𝑛=3
𝑛=4
𝑛=1
0, , , ,…
How are these terms related to each other? Can they be defined in a similar way?
Can the terms can be re-written using their relative positions (n values) ?
The next term would be 4(5)=20
To write a rule, relate the terms to their relative position (n values).
0(1),1(2),2(3),3(4)
𝑎 𝑛 =
(𝑛−1)(𝑛)

6. Solving a Real-Life Problem
** Do not copy, just pay attention.

7. Writing Rules for Series
When the terms of a sequence are added together, the resulting expression is a series. A series can be finite or infinite.
Finite Series:
Infinite Series: …
𝑖=1 4 2𝑖
Summation Notation/Sigma Notation:
𝑖=1 ∞ 2𝑖
Summation Notation/Sigma Notation:
𝑖 is the index (like the relative positon/n-value) and the lower limit of the summation.
4 is the upper limit in the finite series, and ∞ is the upper limit in the infinite series.

8. Writing Series Using Summation Notation
Example 4: Write each series using summation notation.
a) ⋯ b) ⋯
** try to re-write the terms using their index (relative position) ⋯
⋯25(10)
Ending point
𝑖= 𝑖
Ending point
𝑖=1 ∞ 𝑖 𝑖+1
Starting point
Starting point

9. Finding the Sum of a Series
Note: The index of a summation does not have to be 𝑖, it could be any variable, and it does not have to start at 1.
Example 5: Find the sum
𝑘=4 8 (3+ 𝑘 2 )
=( )
+( )
+( )
+( )
+( ) =
For series with many terms, finding the sum by adding the terms can be time consuming. There are formulas for special types of series.
=205

10. Formulas For Special Series
𝑖 𝑛 1 =𝑛
Sum of n terms of 1:
Sum of first n positive integers:
Sum of squares of first n positive integers:
𝑖 𝑛 𝑖 = 𝑛(𝑛+1) 2
𝑖 𝑛 𝑖 2 = 𝑛(𝑛+1)(2𝑛+1) 6

11. Using a Formula For a Sum
𝑖 𝑛 1 =𝑛
𝑖 𝑛 𝑖 = 𝑛(𝑛+1) 2
𝑖 𝑛 𝑖 2 = 𝑛(𝑛+1)(2𝑛+1) 6
Example 6: How many apples are in the stack in example 3?
The series in example 3 is given by the formula 𝑎 𝑛 = 𝑛 2 and 𝑛=1,2,3,…,7
𝑖=1 7 𝑖 2
= 𝑛(𝑛+1)(2𝑛+1) 6
= 7(7+1)(2(7)+1) 6
= 7(8)(15) 6
=140
There are 140 apples in the stack.

12. Monitoring Progress Write the first six terms of the sequence.
1) 𝑎 𝑛 =𝑛+4 2) 𝑓 𝑛 = (−2) 𝑛−1 3) 𝑎 𝑛 = 𝑛 𝑛+1
Find the pattern, write the next term, and write a rule for the nth term of the sequence.
4) 3,5,7,9,…
5) 3,8,15,24,…
6) 1,−2,4,−8,…
7) 2,5,10,17,…
5,6,7,8,9,10
1,−2,4,−8,16,−32
1 2 , 2 3 , 3 4 , 4 5 , 5 6 , 6 7
2+1, 4+1, 6+1, 8+1,…
11; 𝑎 𝑛 =2𝑛+1
1(3), 2 4 , 3 5 , 4 6 …
35; 𝑎 𝑛 =𝑛(𝑛+2)
(−2) 0 , (−2) 1 , (−2) 2 , (−2) 3
16; 𝑎 𝑛 = (−2) 𝑛−1
26; 𝑎 𝑛 = 𝑛 2 +1
1+1, 4+1, 9+1, 16+1,…

13. Monitoring Progress Write the series using summation notation.
9) … ) ⋯
11) ⋯ 12) …+12
Find the sum.
13) 14) 15) 16) ⋯
𝑖=1 20 5𝑖
𝑖=1 ∞ 𝑖 2 𝑖 2 +1
5(1)+5(2)+5(3)+…+5(20)
𝑖=1 ∞ 6 𝑖 …
(3+4)…+(8+4)
𝑖=1 8 (𝑖+4)
𝑖=1 5 8𝑖
𝑘=3 7 ( 𝑘 2 −1)
𝑖=1 34 1
𝑘=1 6 𝑘
=120
=130
=34
=21