Copyright © 2011 Pearson Education, Inc. Slide 11.1-1 Obj: The student will demonstrate the ability...
Transcript of Copyright © 2011 Pearson Education, Inc. Slide 11.1-1 Obj: The student will demonstrate the ability...
Copyright © 2011 Pearson Education, Inc. Slide 11.1-1
Obj: The student will demonstrate the ability to evaluate the first five terms of explicit and recursive sequences.
DrillWhat is the next shape/number for each?1.
2.5, 3, 1, -1, -3, ____3.1, 4, 9, 16, 25, ____4.2, 4, 8, 16, 32, ____
Copyright © 2011 Pearson Education, Inc. Slide 11.1-2
11.1 Sequences
Sequences are ordered lists generated by a
function, for example f(n) = 100n
(1), (2), (3),...
100,200,300,...
f f f
Copyright © 2011 Pearson Education, Inc. Slide 11.1-3
• f (x) notation is not used for sequences.• Write • Sequences are written as ordered lists
• a1 is the first element, a2 the second element, and so on
11.1 Sequences
A sequence is a function that has a set of natural numbers (positive integers) as its domain.
( )na f n
1 2 3, , , ...a a a
Copyright © 2011 Pearson Education, Inc. Slide 11.1-4
11.1 Graphing Sequences
The graph of a sequence, an, is the graph of thediscrete points (n, an) for n = 1, 2, 3, …
Example Graph the sequence an = 2n.
Solution
Copyright © 2011 Pearson Education, Inc. Slide 11.1-5
11.1 Sequences
A sequence is often specified by giving a formula forthe general term or nth term, an.
Example Find the first four terms for the sequence
Solution
1
2n
na
n
1 2(1 1) /(1 2) 2 / 3, (2 1) /(2 2) 3 / 4a a
3 4(3 1) /(3 2) 4 / 5, (4 1) /(4 2) 5 / 6a a
Copyright © 2011 Pearson Education, Inc. Slide 11.1-6
11.1 Sequences
• A finite sequence has domain the finite set
{1, 2, 3, …, n} for some natural number n.
Example 1, 2, 3, 4, 5, 6, 7, 8, 9, 10
• An infinite sequence has domain
{1, 2, 3, …}, the set of all natural numbers.
Example 1, 2, 4, 8, 16, 32, …
Copyright © 2011 Pearson Education, Inc. Slide 11.1-7
11.1 Convergent and Divergent Sequences
• A convergent sequence is one whose terms get closer and closer to a some real number. The sequence is said to converge to that number.
• A sequence that is not convergent is said to be divergent.
Copyright © 2011 Pearson Education, Inc. Slide 11.1-8
11.1 Convergent and Divergent Sequences
Example The sequence converges to 0.
The terms of the sequence 1, 0.5, 0.33.., 0.25, … grow smaller and smaller approaching 0. This can be seen graphically.
1na
n
Copyright © 2011 Pearson Education, Inc. Slide 11.1-9
11.1 Convergent and Divergent Sequences
Example The sequence is divergent.
The terms grow large without bound
1, 4, 9, 16, 25, 36, 49, 64, …
and do not approach any one number.
2na n
Copyright © 2011 Pearson Education, Inc. Slide 11.1-10
n
n
na2
1
Replacing n with n = 1, 2, 3, 4, and 5 will give you the first five terms.
Copyright © 2011 Pearson Education, Inc. Slide 11.1-11
11.1 Sequences and Recursion Formulas
• A recursion formula or recursive definition defines a sequence by– Specifying the first few terms of the sequence
– Using a formula to specify subsequent terms in terms of preceding terms.
Copyright © 2011 Pearson Education, Inc. Slide 11.1-12
11.1 Using a Recursion Formula
Example Find the first four terms of the sequence a1 = 4; for n >1, an = 2an-1 + 1
Solution We know a1 = 4.
Since an = 2an-1 + 1
2 1
3 2
4 3
2 1 2 4 1 9
2 1 2 9 1 19
2 1 2 19 1 39
a a
a a
a a
Copyright © 2011 Pearson Education, Inc. Slide 11.1-13
11.1 Applications of Sequences
Example The winter moth population in thousandsper acre in year n, is modeled by
for n > 2
(a) Give a table of values for n = 1, 2, 3, …, 10
(b) Graph the sequence.
21 1 11, 2.85 0.19n n na a a a
Copyright © 2011 Pearson Education, Inc. Slide 11.1-14
11.1 Applications of Sequences
Solution(a)
(b)Note the population stabilizes near a value of 9.7 thousand insects per acre.
n 1 2 3 4 5 6
an 1 2.66 6.24 10.4 9.11 10.2
n 7 8 9 10
an 9.31 10.1 9.43 9.98
Copyright © 2011 Pearson Education, Inc. Slide 11.1-15
Class workName:____________
Write the first five terms of each sequence.(explicit)1. an = 2n + 3 2. an = n3 + 1
1. an = 3(2n ) 4. an = (-1)n (n)
Find the third, fourth and fifth terms of each.(recursive)3. a1 = 6; an = an-1 + 4 6. a1 = 1; an = an-1 + 2n – 1
7. a1 = 9; an = an-1 8. a1 = 4; an = (an-1 )2 - 10
3
1
Copyright © 2011 Pearson Education, Inc. Slide 11.1-16
11.1 Series and Summation Notation
• Sn is the sum a1 + a2 + …+ an of the first n terms of the sequence a1, a2, a3, … .
is the Greek letter sigma and indicates a sum.
• The sigma notation means add the terms ai
beginning with the 1st term and ending with the nth term.
• i is called the index of summation.
1
n
ii
a
Copyright © 2011 Pearson Education, Inc. Slide 11.1-17
11.1 Series and Summation Notation
A finite series is an expression of the form
and an infinite series is an expression of the form
.
1 2 31
...n
n n ii
S a a a a a
1 2 31
... ...n ii
S a a a a a
Copyright © 2011 Pearson Education, Inc. Slide 11.1-18
11.1 Series and Summation Notation
Example Evaluate
(a) (b)
Solution(a)
(b)
6
1
(2 1)k
k
6
3j
j
a
61 2 3 4
1
5 6
(2 1) (2 1) (2 1) (2 1) (2 1)
(2 1) (2 1)
3 5 9 17 33 65 132
k
k
6
3 4 5 63
jj
a a a a a
Copyright © 2011 Pearson Education, Inc. Slide 11.1-19
Examples of Finite Series
1.
2.
3.
6
2k
k
7
3
74n
n
5
1
4k
k
Copyright © 2011 Pearson Education, Inc. Slide 11.1-20
11.1 Series and Summation Notation
Summation Properties
If a1, a2, a3, …, an and b1, b2, b3, …, bn are two sequences, and c is a constant, then, for every positive integer n,
(a) (b)
(c)
1
n
i
c nc
1 1
n n
i ii i
ca c a
1 1 1
( )n n n
i i i ii i i
a b a b
Copyright © 2011 Pearson Education, Inc. Slide 11.1-21
11.1 Series and Summation Notation
Summation Rules
1
2 2 2 2
1
2 23 3 3 3
1
( 1)1 2 ...
2
( 1)(2 1)1 2 ...
6
( 1)1 2 ...
4
n
i
n
i
n
i
n ni n
n n ni n
n ni n
Copyright © 2011 Pearson Education, Inc. Slide 11.1-22
11.1 Series and Summation Notation
Example Use the summation properties to
evaluate (a) (b) (c)
Solution
(a)
40
1
5i
22
1
2i
i
142
1
(2 3)i
i
40
1
5 40(5) 200i