Ch 8.1 Sequences and Series
Date:_______________
Essential Question: How do I represent a sequence of numbers or the sum of a sequence?
Topic/Question
Notes
Warm Up: Finding
Patterns
What are
Sequences?
What pattern do you notice in the following sequences of numbers? For each of the
following, describe the pattern, write the next term, and write a rule for the nth
term.
2 4 6 8
a) -1, -8, -27, -64…
d) 5, 5, 5, 5
b) 0, 2, 6, 12….
e) 25, 50, 75, 100, 125…
c) -3, -1, 1, 3…
f)
A sequence is a collection of items in a specific order so that it has a first member, a
second member, a third member, and so on.
Mathematically, it is a function whose domain is ____________________________
An infinite sequence __________________________________________________
EX 1: Writing the
terms of a
sequence
A finite sequence _____________________________________________________
Write the first four terms of each sequence
𝑎𝑛 = 3 + (−1)𝑛
c) 𝑎𝑛 =
1
2𝑛
(−1)n
B) 𝑎𝑛 = 2n −1
e) 𝑎𝑛 =
2𝑛
𝑛!
6
d) 𝑎𝑛 = (𝑛+1)(𝑛2 −𝑛+6)
Is it sufficient to define a unique system by just listing its first few terms? Why?
Some sequences
are defined
recursively.
To define a sequence recursively, you need to be given one or more of the first few
terms. All other terms of the sequence are then defined using previous terms.
EX 2: Finding the
nth term of a
sequence
a) 5, 8, 11, 14, …
The ____________
Sequence
0, 1, 1, 2, 3, 5, 8, 13, 21, 34….
The pattern in words: _____________________________________________
Defined recursively as follows: 𝑎0 = 1, 𝑎1 = 1, 𝑎𝑘 = 𝑎𝑘−2 + 𝑎𝑘−1 where k ≥ 2
𝑎0 = 1
𝑎1 = 1
𝑎2 =
𝑎3 =
𝑎4 =
b) 4, 6, 8, 10, 12, …
i = _________________
Summation
Notation
You Try!
n = _________________
The sum of the first n terms of the sequence is called a finite series
The sum of all terms of the infinite sequence is called an infinite series
5
∑ 3𝑖
𝑖=1
6
∑( 1 + 𝑘 2 )
𝑘=3
8
∑
𝑛=0
Some Properties
Reflection
1
𝑛!