Transcript Document

SFM Productions Presents:
Another action-packet episode of
“Adventures inPre-Calculus!”
9.1
Sequences and Series
Homework for section 9.1
p647
Part 1: 19-35, 47-73 (EOO if too much)
Part 2: 77-119 (EOO if too much)
A sequence is a function whose domain is the set of
positive integers.
Written as:
a 1,
w here a 1 means: the f irst t erm.
a2 ,
w here a 2 means: the second term.
. a3,
.
.
a n,
w here a 3 means: the third term.
w here a n means: the n th t erm.
I nf inite S equence:
domain is set of + integers.
Finite Sequence: domain consists of the first n
integers.
positive
Sometimes can start with a subscript of 0…
a0, a1, a2., a3…
Finding terms of a sequence:
a n  3   1
f irst f ive terms are:
an  5 n  3
f irst f ive terms are: 2 , 7, 12 , 17, 2 2
n
2n
 n
3
f irst f ive terms are:
2 4 8
16
32
, ,
,
,
3 9 2 7 8 1 2 43
 1 f irst f ive terms are:

2n 1
1
1 1
1
1,
, 
, , 
3
5 7
9
an
n
an
2 , 4, 2 , 4, 2
There may be more than one pattern that works, but we are only after the most apparent term…
1 1 1 vvvv
, ,
,
2 4 8
1
16
1
15
Find the nth term:
an
1
 n
2
an 
6
n  1 n 2  n  6 
(that means find the formula, or model……
an = some formula)
1, 3 , 5 , 7 ...
an  2 n  1
n
1
2
3
4
an
1
3
5
7
2 , 5 , 10 , 17...
n
an
1
2
a n  n2  1
2
3
4
5
10
17
3 , 7, 11, 15 , 19 ...
n
an
1
3
2
7
3
11
4
15
5
19
a n  4n  1
1 2
4
8
16
,
,
,
,
...
3 9 2 7 8 1 2 43
n
1
an
1
3
2
2
9
3
4
27
4
5
8
16
8 1 2 43
an
2 n 1

3n
Some sequences are defined recursively, which means you need to be given one or more of the first
term(s) - then the following terms can be found using the previous ones.
Recursive sequences use: ak
Given:
a0  1
Find the next 4 terms…
a1  1
and a k  a k 2  a k 1
Find the next 4 terms…
Given:
a0  1
a0 
vvvv
1
a1 
1
and a k  a k 2  a k 1
a1  1
We want a2. And since recursive sequences have the form: ak, that
means k = 2.
a2 
vvvv
2
a 2  a 2 2  a 2 1  a 0  a 1  1  1  2
a3 
vvvv
a 3  a 3 2  a 3 1  a 1  a 2  1  2
3
a4 
vvvv
a 4  a 4 2  a 4  1  a 2  a 3  2  3
5
 5
a5 
vvvv
8
a 5  a5
 8
2
 a5
1
 a3  a4  3  5
You’re set to do up through problem 59.
 3
Another type of sequence is defined as: !
! = factorial
If n is an integer, then n! is defined as:
n !  n  n  1  n  2
  n
 3   ...  3  2  1
B y def init ion:
0 !
1!
2 !
3 !
4!
5 !






1
1
2
3
4
5
1 2
2 1 6
3 2 1  24
 4  3  2  1  12 0
Evaluating factorials:
8 !
8 7
8 7 6 5 4 3 2 1
 28


2 !6 !
2 1
2  16 5 4  3  2  1
2 !6 !
3 !5 !

2  16 5 4  3  2  1
3  2  15 4  3  2  1

6
3
 2
n  n  1  n  2   ...  3  2  1
n!
 n

n  1 !
n  1  n  2   n  3   ...  3  2  1
You’re set to do up through problem 83.
Summation Notation:
a convenient way to notate the sum of the terms of a
finite sequence.
Also known as Sigma notation.

T he sum of the first n terms of a sequence:
n
ai

i 1
 a 1  a 2  a 3  ...  a n
i = index of summation
1 = low er limit
n = upper limit
Find the sum of the first 6 terms (starting with 1)
of the sequence: 3n - 1.
3 1  1  3 2   1  3 3 
3 5   1  3 6   1 
6
 3 i
 1 
vvvv
i 1
 1  3 4   1 
3  1  6  1  9  1  12  1  15  1  18  1 
2 
 5   8

 11  14   17  
5 7
Break out your battery operated brain…
5
2 i
2
 110
i 0
10

i 0
 1
   2 .7 18 2 8 18 2 8 46 .........
i ! 
 e
Properties of Sums of sequences:
Finite series (also called the nth partial sum)
Infinite series

3

i
i 1 10
1. Find the 3rd partial sum.
2. Find the sum of the whole thing…

3

i
i 0 10
1. Find the 3rd partial sum.
2. Find the sum of the whole thing…
Go! Do!