13.1: Sequences
Consider the following list of numbers:
2, 5, 10, 17, 26, 37, …
Can you write a rule for this list? There may be multiple correct
answers.
Def: A sequence is a function whose domain is the set of positive
integers.
For example, the sequence 𝑎𝑛 = 3𝑛 − 1 is a list whose first 5 terms
are: {2, 5, 8, 11, 14}
1.
Find the first five terms of each sequence:
3 𝑛
a. 𝑠𝑛 = ( )
4
b. 𝑝𝑛 =
2𝑛
𝑛2
Notice how the nth term in the sequence is the value of the function at
n. For example, plugging in 20 for n returns the 20th term of the
sequence.
2.
For each sequence, determine 𝑎𝑛 , the nth term of the sequence.
a. 2, 4, 6, 8, 10, …
b. 3, 5, 9, 17, 33, …
1 1
1
3 9
27 81
c. − , , −
,
1
,−
1
243
d. 1, −4, 9, −16, 25, …
e. 1, 2, 6, 24, 120, …
…
3.
Calculate:
a. 6!
b.
c.
10!
8!
12!
4!8!
(𝑛+2)!
d. (𝑛−1)!
Here’s another way we could denote factorial:
𝑎1 = 1;
𝑎𝑛 = 𝑛 ∙ 𝑎𝑛−1
This is called a recursively defined sequence. A recursive sequence will
always have one or more base cases and a rule for calculating each
term based on its predecessor(s).
4.
Write down the first 5 terms of the recursive sequence:
a. 𝑎1 = 8; 𝑎𝑛 = 2 − 𝑎𝑛−1
b. 𝑟1 = 1;
𝑟𝑛 = 𝑟𝑛−1 + 2𝑛 − 1
c. 𝑓1 = 1;
𝑓2 = 1;
𝑓𝑛 = 𝑓𝑛−2 + 𝑓𝑛−1
Sometimes, we don’t want to just find the first n terms in a sequence;
we want to find their sum! It can get a little cumbersome writing down
𝑎1 + 𝑎2 + 𝑎3 + ⋯ + 𝑎𝑛 . Instead, we will often use summation
notation:
𝑛
∑ 𝑎𝑘 = 𝑎1 + 𝑎2 + 𝑎3 + ⋯ + 𝑎𝑛
𝑘=1
5.
Find the sums:
a.
4
∑ 𝑘3 − 3
𝑘=1
b.
5
∑ 2𝑘 2 + 5𝑘
𝑘=3
Some properties:
And some formulas:
6.
Use properties (1)-(8) to calculate the sums:
a.
20
∑ 3𝑘 − 4
𝑘=1
b.
30
∑ 5𝑘 2 + 𝑘
𝑘=9
c.
50
∑ 7 − 𝑘3
𝑘=13
Homework 13.1a:
p. 946: #9-77 EOO, 86-88
Homework 13.1b:
p. 946: #11-79 EOO, 93, 97-99