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!