Download Writing Rules for Series

Section 8.1—Defining and Using Sequences and Series
Writing Terms of Sequences
 Sequences
 A sequence is an ordered list of numbers.
 A finite sequence is a function that has a limited number of terms.
 Example 1: 2, 4, 6, 8
Example 2: 3, 5, 7, 9
 The domain is the set of values being inputted for the function. The domain
usually begins at 1, however there are certain times it may begin at 0.
Unless told differently, assume the domain for a sequence begins at 1.
 The values of the range are called the terms of the sequence. They are the
values you receive when plugging in the values of the domain.
 An infinite sequence is a function that continues without stopping.
 Example 3: 2, 4, 6, 8, . . .
Example 4: 3, 5, 7, 9, . . .
 The domain is the set of all positive real integers.
 If the values of the domain begin with 0 rather than 1, the domain will be the
set of all nonnegative real integers.
 A sequence can be written as an equation or a rule. The rule for both example 1 and
3 could be 𝑎𝑛 = 2𝑛 or 𝑓(𝑛) = 2𝑛.
EXAMPLE 1: Writing the Terms of Sequences
Write the first six terms of:
Part A:
𝑎𝑛 = 2𝑛 + 5
Part B:
𝑓 (𝑛) = (−3)𝑛−1
EXAMPLE 2: Writing Rules for Sequences
Describe the pattern, write the next term, and write a rule for the nth term of the sequences
Part A: -1, -8, -27, -64, . . .
Part B: 0, 2, 6, 12, . . .
EXAMPLE 3: Solving a Real-Life Problem
You work in a grocery store and are stacking apples in the shape of a square pyramid with
seven layers. Write a rule for the number of apples in each layer. Then graph the sequence.
Writing Rules for Series
 Series
 A series is created by adding up the terms of a sequence. A series can be finite or
infinite.
 A series can be written using summation notation (also known as sigma notation).
 Finite Series Example:
Infinite Series Example:
4
2 + 4 + 6 + 8 = ∑ 2𝑖
∞
2 + 4 + 6 + 8 + . . . = ∑ 2𝑖
𝑖=1
𝑖=1
 The index of summation for both is i. The lower limit of summation is 1. The upper
limit of summation is 4 for the finite series and ∞ (infinity) for the infinite series.
EXAMPLE 4: Writing Series Using Summation Notation
Write each series using summation notation.
Part A: 25 + 50 + 75 + . . . + 250
Part B:
1
2
2
3
4
3
4
5
+ + + +. . .
EXAMPLE 5: Finding the Sum of a Series
Find the sum
8
∑(3 + 𝑘 2 )
𝑘=4
EXAMPLE 6: Using a Formula for a Sum
How many apples are in the stack in Example 3?
Monitoring Progress
Write the first six terms of the sequence.
1) 𝑎𝑎 = 𝑛 + 4
3) 𝑎𝑛 =
2) 𝑓 (𝑛) = (−2)𝑛−1
𝑛
𝑛+1
Describe the pattern, write the next term, graph the first five terms, 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, . . .
Write the series using summation notation.
9) 5 + 10 +15 + . . . + 100
10)
1
2
4
9
5
10
+ +
+
16
17
+...
11) 6 + 36 + 216 + 1296 + . . .
12) 5 + 6 + 7 + . . . + 12
Find the sum.
13)
14)
5
7
∑ 8𝑖
∑(𝑘 2 − 1)
𝑖=1
𝑘=3
15)
34
16)
6
∑1
∑𝑘
𝑖=1
𝑘=1