Download 12.1 Sequences

13.1
Sequences
Definition of a Sequence
• 2, 5, 8, 11, 14, …, 3n-1, …
• A sequence is a list.
• A sequence is a function whose domain is the set of natural
numbers. In mathematics, natural numbers are the ordinary
counting numbers 1, 2, 3, ... (sometimes zero is also included).
• The values of this function are called terms.
• Sometimes a sequence is called a progression.
• The notation an  is used to denote the sequence.
Find the next term in each sequence below.
#1
1,3,7,11,13,…
#2
230,460,46,92,9.2,…
#3
3,12,24,33,66,…
• Example
Write the first three terms of the sequence
where an  n2  1 .
• All sequences have an infinite number of
terms. In some cases, we want to consider
only a finite number of terms. An array is a
function whose domain is a finite subset of
the natural numbers.
• Example
Write the array consisting of the first six terms
of the sequence an   5n  3 .
Sequences of Partial Sums
•
•
•
•
also known as “finite series”
Math or engineering fields
Computer science
Vectors in HPC
• A partial sum is the total of the terms of a
sequence up to a given term.
• Associated with every sequence a  is another
sequence S  called the sequence of partial
sums.
n
n
• Example
List the first five terms of the sequence 2n 1 .
Find the first 4 terms of the sequence of
partial sums.
Find the 4th term.
#1
#2
5,25,125,…, 5n ,…
a1  1, an  2an 1  3
n

#3
th
prime #
List the first six terms.
#1 2n  7
#2
#3
2
n
  1
n
th
prime #
Give the fourth through eighth terms of each sequence.
#1
a1  3, an  2an 1
#2
an   n 2  2n
List the first five terms of the sequence an   2n  1 .
Find the first 4 terms of the sequence of partial sums.