Download Warm-up - mathew1

Warm-up
Describe the pattern and find the next three
terms.
1. 2, 3, 5, 8, 12, ....
2. 1/2, 1/4, 1/8, 1/16, 1/32...
3. 5, 2, 6, 3, 9, 6, 14, ....
1
[9.1] SEQUENCES AND SERIES
Sequences
def: a collection of terms ordered so that it
has a first member, second member, etc...
a1, a2, a3, a4, . . . , an
a1 is the first term!
&
an is the nth term!
Ex.) 2, 3, 5, 8, 12, .... is a sequence.
a1 =
a3 =
2
Two Types of Sequences
a) finite sequence
3, 6, 9, 12, 15
# of terms?
b) infinite sequence
3, 6, 9, 12, 15, ....
# of terms?
3
Explicit Form - a formula for a sequence where
each term is found by plugging in
the term number.
To find the first term, simply plug in 1 for n
because a1 is the first term and the value of n is 1.
Ex 1: Write the first five terms of
4
Recursive Sequences: The first term(s) will be given and
all other terms will be defined using previous terms.
In a recursive sequence, the next term is found by using
the term before it.
Example:
Find the first five terms.
1. a1 = 3, ak + 1 = ak - 4
2. a1 = 15, ak + 1 = 2ak + 10
5
*You Try
1. a1 = -2, ak + 1 = ak
Find a3
+
2
2. a1 = 11, ak + 1 = 3ak + 2
Find a4.
6
The most famous recursive sequence is the
Fibonacci Sequence:
Definition:
a0 = 1, a1 = 1, ak = ak-2 + ak-1 , where k≥2
Terms:
a0 = 1
a1 = 1
a2 = a2-2 + a2-1= a0 + a1 = 1 + 1 = 2
a3 = a3-2 + a3-1= a1 + a2 = 1 + 2 = 3
a4 = a4-2 + a4-1= a2 + a3 = 2 + 3 = 5
Find a5 and a6.
7
*You Try:
Determine if the sequence is explicit form or recursive.
Then, write the first 5 terms of the sequence.
1. a1 = 4, ak +1 = ak + 5
2. an = 3(-5)n-1
3. a1 = -4, ak +1 = 3(ak + 2)
8
Series
def: the sum of the terms of a sequence
a) finite
b) infinite
3 + 6 + 9 + 12 + 15
3 + 6 + 9 + 12 + 15 + ....
9
Sigma (or summation) Notation
=
10
Ex : Find the sum of the series.
a)
b)
11
Factorials:
0! = 1
1! = 1
2! = 2 1 = 2
3! = 3 2 1 = 6
7! = 7 6 5 4 3 2 1 = 5040
12
Evaluating factorials:
1.
8!
2!
6!
2. 42! 52!
45! 50!
3. (n + 2)!
n!
13
[9.2] Arithmetic Sequences
EQ: How do we write a rule for
an arithmetic sequence?
I. Definition of Arithmetic Sequence: a
sequence where the difference between terms
is constant
*In other words, an arithmetic sequence is a sequence where
you add the same number to each term to get the next term.
Ex 1: -3, 1, 5, 9, 13, ...
What is the difference between the numbers?
common difference is d =
Ex 2: 2, -5, -12, -19, ...
14
*You try:
Determine if the sequence is arithmetic.
And, if it is, find the common difference, d.
1.) -10, -6, -2, 0, 2, 6, 10
2.) 5, 11, 17, 23, 29
15
Formula for an Arithmetic Sequence
an = dn + c
Where d is the common difference and c = a1 - d
Notice that arithmetic sequences are linear!
16
Ex 3: In an arithmetic sequence the common
difference is 5 and a1 = -3.
Write the formula for the sequence and find a15.
Ex 4: Write a formula for the arithmetic sequence
where a5 = 25 and a12 = 39
17
*You Try:
1.) In an arithmetic sequence the common
difference is -2 and a1 = 7.
Write the formula for the sequence and find a20.
2.) Write a formula for the arithmetic sequence
where a12 = -50 and a20 = -18.
18
EQ: How do you find the sum of an
arithmetic series?
The Sum of a Finite ARITHMETIC Series
The sum of the first n terms of an arithmetic
series is:
Which of these is arithmetic? How do you know?
27
Σ
k=1
27
2k - 3
Σ
(2k - 3)2
k=1
19
Ex.)
Find the sum of the first 25 terms of the arithmetic series
(this is also called the 25th partial sum of the series):
20 + 18 + 16 + 14 + ...
1. Find an using an = dn + c
2. Then, plug into
*You try:
Find the 30th partial sum of the following series:
11 + 16 + 21 + 26 + ...
20
What if the series is written in sigma notation?
Ex: Find the sum of the following series.
*You try:
What about something like these?
7
Σ
k=3
7
-2k + 1
Σ
-2k2 + 1
k=3
21
Homework:
Pg. 625
#s 1,4,9,26,30,69-84 mult. of 3
Pg. 635
#s 1-8,35,40,44,60,61,68,71
22