SFM Productions Presents: Another action-packet episode of “Adventures inPre-Calculus!”...
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Transcript of SFM Productions Presents: Another action-packet episode of “Adventures inPre-Calculus!”...
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.
11
Written as:
where a means: the first t ., erma
22 where a means: the second , term.a
33 where a means: the third , term.a...
thnwhere a means: the n t, m er .na
I nfinite S equence: domain is set of + integers.
Finite Sequence: domain consists of the first n positive integers.
Sometimes can start with a subscript of 0…a0, a1, a2., a3…
Finding terms of a sequence:
3 1nna first five terms are: 2 , 4, 2 , 4, 2
5 3na n first five terms are: 2 , 7, 12 , 17, 22
23
n
n na first five terms are:2 4 8 16 32
, , , ,3 9 27 81 243
1
2 1
n
na nfirst five terms are:
1 1 1 11, , , ,
3 5 7 9
There may be more than one pattern that works, but we are only after the most apparent term…
1 1 1, , ,
2 4 8vvvv 1
16
12n na
115
2
6
1 6na
n n n
Find the nth term: (that means find the formula, or model……
an = some formula)1, 3, 5 , 7...
n
an
1 2 3 4
1 3 5 7
2 1na n
2, 5 , 10, 17...
n
an
1 2 3 4
2 5 10 17
2 1na n
3, 7, 11, 15 , 19...
n
an
1 2 3 4 5
3 7 11 15 19 4 1na n
1 2 4 8 16, , , , ...
3 9 27 81 243
n
an
1 2 3 4 5
1 2 4 8 163 9 27 81 243
12
3
n
n na
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:
0 11 1a a 12and k kka a a
Find the next 4 terms…
Given:
0 11 1a a 12and k kka a a
Find the next 4 terms…
0
1
2
3
4
5
1
1
vvvv
vvvv
vvvv
vvv
a
a
a
v
a
a
avvvv
We want a2. And since recursive sequences have the form: ak, that means k = 2.
22 2 2 1a a a 0 1a a 1 1 22
3a 23 13a a 1 2a a 1 2 33
4a 24 14a a 32a a 2 3 55
5a 25 15a a 3 4a a 3 5 88
You’re set to do up through problem 59.
Another type of sequence is defined as: !
! = factorial If n is an integer, then n! is defined as:
! 1 2 3 ... 3 2 1n n n n n
B y definiton:
0 !
i
1
1! 1 2 ! 2 1 2
3 ! 3 2 1 6
4 ! 4 3 2 1 24 5 ! 5 4 3 2 1 120
Evaluating factorials:
8 !2 ! 6 !
8 7 6 5 4 3 2 12 1 6 5 4 3 2 1
8 72 1
28
2 ! 6 !3 ! 5 !
2 1 6 5 4 3 2 13 2 1 5 4 3 2 1
63
2
!1 !
nn
1 2 ... 3 2 1
1 2 3 ... 3 2 1
n n n
n n n n
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:
i = index of summation
1 = lower limit
n = upper limit
1 32
1...
n
niia a a a a
Find the sum of the first 6 terms (starting with 1) of the sequence: 3n - 1.
6
1
3 1i
ivvvv
3 1 1 3 2 1 3 3 1 3 4 1
3 5 1 3 6 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
0
2i
i 110
10
0
1!i i
2 .71828182846.........
e
Properties of Sums of sequences:
Infinite series
Finite series (also called the nth partial sum)
1
310 i
i
1. Find the 3rd partial sum.
2. Find the sum of the whole thing…
0
310 i
i
1. Find the 3rd partial sum.
2. Find the sum of the whole thing…
Go! Do!