Download A sequence is a function whose domain is a set of consecutive

12.1 - Sequences and Series
Tuesday, April 06, 2010
9:21 AM
A sequence is a function whose domain is a set of consecutive integers.
Finite Sequences
Infinite Sequences
2, 4, 6, 8
2, 4, 6, 8 . . .
3, -6, 9, -12, 15
3, -6, 9, -12, 15 . . .
Ex 1: Write terms of a sequence
Write the first 6 terms of the following sequences:
2.
1.
Write the first 6 terms of the sequences below.
1.
2.
Ch 12 Page 1
3.
Example 2: Write an equation that describes the pattern of the following:
1. -1, -8, -27, -64, . . .
3. 3,
8,
15,
2.
0, 2, 6, 12, . . .
24, . . .
Series and Summation Notation
When the terms of a sequence are added together, the result is a series.
Finite series: 2 + 4 + 6 + 8
Infinite series: 2 + 4 + 6 + 8 + . . .
You can use summation notation (sigma notation) to write a series.
Example 4: Write the series using summation notation.
1. 25 + 50 + 75 + . . . + 250
Ch 12 Page 2
2.
Guided Practice for Ex4
Write the series using summation notation.
1.
5 + 10 + 15 + . . . + 100
2.
6 + 36 + 216 + 1296 + . . .
3.
5 + 6 + 7 + . . . + 12
Find the sum of a series.
Ch 12 Page 3
12.2 - Arithmetic Sequences and Series
Saturday, April 03, 2010
4:24 PM
In an arithmetic sequence, the difference between the terms is constant and is called
the common difference. (d)
Example 1: Identifying arithmetic sequences
Is the sequence arithmetic? If yes, what is d?
1) -4, 1, 6, 11, 16, …
2) 3, 5, 9, 15, 23, …
3) 17, 14, 11, 8, ...
The RULE for an arithmetic sequence:
For example: Write the rule for the nth term of an arithmetic sequence with a first
term of 2 and a common difference of 3.
Example 2: Write a rule for the nth term, then find
1. 4, 9, 14, 19 . . .
2. 60, 52, 44, 36 . . .
Example 3: Write a rule when you are given one term and d.
One term of an arithmetic sequence is
= 48 and d = 3
Ch 12 Page 4
Extra Ex 3: Write a rule for the nth term
One term of an arithmetic seq is
= 263 and d = 11.
Example 4: Write a rule given 2 terms
Write a rule for the nth term.
Extra Ex4: Write a rule for the nth term.
Ch 12 Page 5
An arithmetic series is the expression formed by adding the terms of an arithmetic
sequence. The sum of the first terms is denoted by Sn.
1st term
number of terms
last term
Ex 5: What is the sum of the arithmetic series:
Extra ex 5: What is the sum of
Ch 12 Page 6
12.3 Geometric Sequences and Series
Monday, April 12, 2010
11:26 AM
In a geometric sequence, the ratio of any term to the previous term is a constant.
This constant ratio is called the common ratio,
Ex 1: Identify geometric sequences:
Tell whether the sequence is geometric, if yes, find r.
a. 4, 10, 18, 28, 40…
b. 625, 125, 25, 5, 1…
c. 81, 27, 9, 3, 1…
d. 1, 2, 6, 24, 120...
Rule for geometric sequence: The nth term of a geometric sequence with first term
a1 and common ratio r is:
Example: The nth term of a geometric sequence with a first term of 3 and common
ratio 2 is given by:
Ex 2: write a rule for the nth term for the geometric sequence
Write a rule for the nth term of the sequence. Then find a7.
a. 4, 20, 100, 500…
b. 152, -76, 38, -19…
Ch 12 Page 7
c. 3, 12, 48, 192…
d. 36, -12, 4, -4/3…
Ex 3: Write a rule given a term and common ratio
One term of a geometric sequence is
. The common ratio is
a. write a rule for the
term
Extra ex3: One term of a geometric sequence is
Write a rule for the
term
Ex 4: Write a rule given two terms
Two terms of a geometric sequence are
the
term.
. The common ratio is
and
Extra ex 4: Two terms of a geometric sequence are
rule for the
term.
Ch 12 Page 8
.
. Find a rule for
and
. Find a
.
Geometric series: the expression formed by adding the terms of a geometric
sequence. The sum of the first n terms of a series, Sn:
The sum of a finite Geometric Series
= the # of terms you are adding
Ex 5: Find the sum of a geometric series
Ch 12 Page 9
Review for Ch 12 Quiz
Wednesday, May 05, 2010
1:22 PM
1. Find the firsts 4 terms of the sequence
2. a) Write the next 3 terms of the arithmetic sequence. b) Then write the formula
and c) find the 13th term. -10, -7, -4, -1 . . .
3. Find the sum of the series:
4. a) Find the next 3 terms of the geometric sequence. b) Then write the formula
and c) find the 10th term. 3, -12, 48, . . .
Ch 12 Page 10
5. What is the first term of an arithmetic sequence with
.
6. Find the sum of the first 12 terms of the arithmetic series 2 + 6 + 10 + . . .
7. Find the sum of
Leave your answer as a reduced fraction or decimal.
8. Write a rule for the nth term of the geometric sequence with
. Then find .
9. Write a rule for the nth term of the arithmetic or geometric sequence. Find a10, then
Ch 12 Page 11
9. Write a rule for the nth term of the arithmetic or geometric sequence. Find a10, then
find the sum of the first 10 terms of the sequence.
-5, -1, 3, 7 . . .
10. Write a rule for the
a5 = 11 and a11 = 47
term of the arithmetic sequence that has the terms
11. Find the sum of the arithmetic series
Ch 12 Page 12
12.4 - Find the Sum of Infinite Geometric Series
Saturday, April 17, 2010
8:53 PM
Sums of Infinite Geometric Series
Consider the series
Sum of an Infinite Geometric Series =
If
, the series has no sum. So if the numbers are getting bigger, regardless of the sign,
then there is no sum.
Ex 2: Finding sums of infinite geometric series.
What is the sum of the infinite geometric series?
1. 1 - 3 + 9 - 27 + . . .
Ch 12 Page 13
Writing Repeating Decimals as Fractions
• Numerator is the number that repeats
• In the denominator are as many 9's as
numbers that repeat (since 2 numbers
repeat then there are two 9's)
Write the repeating decimal as a fraction in lowest terms.
1. 0.555…
2. 3.727272...
Ch 12 Page 14
3. 0.1531531531...
12.5 - Recursive Rules with Sequences and
Functions
Wednesday, August 17, 2011
2:30 PM
the previous term
Evaluate Recursive Rules
Write the first 6 terms of the sequence.
1.
2.
Recursive Equations for Arithmetic and Geometric Sequences
Arithmetic Sequence
where d is the common difference
Geometric Sequence
where r is the common ratio
Write a recursive rule for the sequence.
1. 3, 13, 23, 33, 43, . . .
2. 16, 40, 100, 250, 625, . . .
Write the first five terms of the sequence.
Ch 12 Page 15
Write the first five terms of the sequence.
Write a recursive rule for the sequence.
1. 2, 14, 98, 686, 4802 . . .
2. 19, 13, 7, 1, -5 . . .
3. 11, 22, 33, 44, 55 . . .
4. 324, 108, 36, 12, 4 . . .
Writing recursive rules for special sequences.
1. 1, 1, 2, 3, 5 . . .
2. 1, 1, 2, 6, 24 . . .
Ch 12 Page 16
3. 1, 2, 2, 4, 8, 32
Test review
Thursday, April 14, 2011
1:49 PM
Write the recursive rule for the sequence. The sequence may be arithmetic,
geometric, or neither.
1.
2.
3. Write the repeating decimal
as a fraction.
4. Write the first five terms of the sequence
where
and
Find the sum of the infinite geometric series, if it exists.
6.
5.
7. Tell whether the sequence
neither.
Ch 12 Page 17
is arithmetic, geometric, or
8. Write a rule for the
. What is
term of the geometric sequence with
and
?
9. Find the sum of the geometric series
10. Write a rule for the
What is
?
term of the arithmetic sequence
11. Write a rule for the
and
term of the arithmetic sequence that has the terms
12. Find the sum of the arithmetic series
Ch 12 Page 18
.
Write the first four terms of the sequence.
13.
14.
15. Find the sum of the series
16. Tell whether the sequence
geometric, or neither.
17. Write the series
notation).
Ch 12 Page 19
is arithmetic,
using summation notation (sigma