Download Section 11.1 Sequences and Summation Notation

Section 11.1
Sequences and Summation Notation
Objectives:
•Definition and notation of sequences
•Recursively defined sequences
•Partial sums, including summation
notation
Sequences
A sequence is a set of numbers written in a
specific order:
a1, a2, a3, a4, …, an, …
– The number a1 is called the first term, a2 is the
second term, and in general an is the nth term.
– Since for every natural number n, there is
a corresponding number an, we can define
a sequence as a function.
Definition of a Sequence
• A sequence is a function f whose domain
is the set of natural numbers.
• The values f(1), f(2), f(3), . . . are called
the terms of the sequence.
Ex 1. Find the first 5 terms and the 100th term of
the sequence defined by each formula.
(a) an  2n  1
(b) cn  n  1
2
Class Work
Find the first 5 terms and the 100th term of the
sequence defined by each formula.
n
1) t n 
n 1
( 1)n
2) rn  n
2
Recursively Defined Sequences
Some sequences do not have simple
defining formulas like those
of the preceding example.
– The nth term of a sequence may depend on some
or all of the terms preceding it.
– A sequence defined in this way is called recursive.
Ex 2. Find the first 5 terms of the sequence
defined recursively by a1 = 1 and an = 3(an–1 + 2).
Class Work
3. Find the first 6 terms of the sequence defined
recursively by a1= 3, a2 = 8 and an = an-1 + an-2.
The Partial Sums of a Sequence
For the sequence
a1, a2, a3, a4, …, an, …
the partial sums are:
S1 = a
S2 = a1 + a2
S3 = a1 + a2 + a3
S4 = a1 + a2 + a3 + a4
Sn = a1 + a2 + a3 + … + an
Ex 3. Find the first four partial sums of the
1
sequence given by n .
2
Class Work
4. Find the first four partial sums of the
sequence given by an  2n  1 .
Sigma Notation
• Given a sequence
a1, a2, a3, a4,…
we can write the sum of the first n terms
using summation notation, or sigma notation.
– This notation derives its name from the Greek
letter Σ (capital sigma, corresponding to our S for
“sum”).
• Sigma notation is used as follows:
n
a
k 1
k
 a1  a2  a3  a4 
 an
– The left side of this expression is read:
“The sum of ak from k = 1 to k = n.”
– The letter k is called the index of summation,
or the summation variable.
Ex 4. Find each sum.
5
(a)  k
k 1
5
2
1
(b) 
j 3 j
Class Work
Find each sum.
10
5)  i
i 5
6
6)  2
i 1
HW #1 Worksheet Sec 11.1