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Section 11.1 ­­ Sequences and Summation Notation
A sequence is a set of numbers written in a specific order:
a1, a2, a3,.......,an,.....
The number a1 is called the first term, a2 is the second, etc.
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
Finding the terms of a sequence
Find the first five terms and the 50th term of the sequence.
an = n2 ­ 3
an = (­1)n(3n ­ 1)
Recursive Sequences
A recursive sequence is one where the nth term may depend on some or all of the preceding terms.
Example: Find the first five term of the recursive sequence defined by an = 3an­1 + 2, = 5
Partial Sums of a Sequence
For the sequence a1, a2, a3,...an,... the partial sums are:
S1 = a1
S2 = a1+ a2
S3 = a1 + a2 + a3
.
.
.
Sn = a1+ a2 + a3 + a4 + ....+ an
S1 is called the first partial sum, S2 is second partial sum and
so on. The sequence S1, S2, S3,...., Sn is called the sequence
of partial sums.
Example: Find the first four partial sums of the sequence given by an = 3n ­ 4
Sigma Notation ­­ the sum of the first n terms can be written
using summation notation or sigma notation.
n
ak = a1 + a2 + a3 +.....+ an
k = 1
The left side is read "the sum of ak from k = 1 to k = n." k is called the index of summation and you need to replace k in the expression after the sigma by the integers 1, 2, 3, 4, ..., n and add the resulting expressions
Example: Find each sum:
5
6
3k
4k2
k = 1
k = 3
Using the calculator to find sums
You will need to type into your calculator the following expression: sum(seq(expression you are finding sum of, variable being used, start value, end value, increment))
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Example: Find the following sum: k = 4
Properties of Sums
n
n
n
(ak + bk) = ak + bk
1. k = 1 k = 1
k = 1
n
n
n
2. (ak ­ bk) = ak ­ bk k = 1
k = 1
k = 1
n
n
3. cak = c ak
k = 1
k = 1
k3 ­ 23