Download 11.1 Intro to Sequences 11.2 Arithmetic and Geometric Sequences

11.1 Intro to Sequences
11.3 Arithmetic and Geometric Means
1. Copy the objective.
2. Define sequence.
3. Define term in a sequence. What is the
mathematical notation for a term in a sequence?
4. How do you know a sequence is a function?
Example: For the following sequence, do the five
things listed in blue on p. 561
5, 8, 11, 14, 17, 20……
What is the 175th term in this sequence?
Example: For the following sequence do the five
things listed on page 561.
2, 6, 18, 54, 162, 486…..
What is the 14th term in this sequence?
1. Copy the objective.
2. Define arithmetic mean in terms of a sequence
3. What is the arithmetic mean of 2 and 8?
4. What does it mean for there to be more than one
arithmetic mean between 2 numbers? Give an
example.
5. What are 2 arithmetic means between 2 and 8?
6. Define geometric mean in terms of a sequence
7. What is one geometric mean between 2 and 8?
8. What does it mean for there to be more than one
geometric mean between 2 numbers? Give example.
9. What are 2 geometric means between 2 and 8?
Example 1.Find 5 arithmetic means between 4 & 36.
Example 2:Find 3 geometric means between 4 &324.
10. What is the special consideration with some sets
of geometric means that is in this example? Explain
11.2 Arithmetic and Geometric Sequences
1. Copy the objective.
2. Define arithmetic sequence in your own words
and give an example
3. Define geometric sequence in your own words and
give an example
Example 1: Is the following sequence arithmetic,
geometric or neither. Explain how you know.
8, 11, 14, 17, 20, ….
Example 2: Is the following sequence arithmetic,
geometric or neither. Explain how you know
64, 32, 16, 8, 4, 2, 1, 1/2……..
Example 3: Is the following sequence arithmetic,
geometric or neither. Explain how you know
5, 12, 22, 35, 51, 70
4. Write the formula for the nth term value of an
arithmetic sequence and explain what the formula
means in English (GES).
5. Write the formula for the nth term in a geometric
sequence and explain what the formula means in
English (GES).
Example 4: Calculate t40 for the arithmetic sequence
3, 10, 17, 24, 31, 38…..
Example 5: Calculate t80 for a geometric sequence
with the first term equal to 21 and the common ratio
r=1.2
Example 6: The number 102 is a term in an
arithmetic sequence with the first term equal to 7 and
the common difference equal to 2.5. Which term is
it?
Example 7: A geometric sequence has the first term
equal to 14 and r equal to 3. If the nth term is
275562 find n.
11.4 Introduction to Series
1. Copy the objective.
2. Define series in your own words Give example.
3. Define nth partial sum of a series in your own
words, and give an example.
4. Why are mathematicians usually interested partial
sums of series rather than the entire series?
5. Explain in English (GES) the following notation
n
t
k 1
k
.
Example: For the following sequence, calculate S5
4, 9, 14, 19, 24, 29, 34, 39, 44, 49…….
11.5 Arithmetic and Geometric Series
1. Copy the objective.
2. Define arithmetic series in your own words and
give an example.
3. Define geometric series in your words give an
example.
4. The blue box formula on p 583 is not very useful.
Copy the blue box at the top of 584.
5. Copy the blue box on page 584 (bottom) - the
partial sum of a geometric series.
Example 1: Find the 86th partial sum of the
arithmetic series with t1=9 and d=6
Example 2: Find S42 for the geometric series with
t1=9 and r= 1.3.
Example 3: 810 is the nth partial sum of the
arithmetic series with first term 11 and common
difference 4. Find n.
Example 4: 99,347.5 is the approximate value of the
partial sum of the geometric series with t1= 70 and
r=1.2. Which partial sum is it?
11.6 Convergent Geometric Series
11.8 Factorials
1. Define asymptote in your own words (GES).
2. Explain in your own words (GES) what it means
for a series to converge.
3. Explain how a convergent series is like an
asymptote (GES).
4. Define limit and write the notation for a limit in
mathematics
5. When does a geometric series converge?
6. Explain what lim n S n means (GES).
7. What is the formula for the value of a convergent
geomtric series?
Example 1: Does the following geometric series
converge?
1 1
32,16,8, 4, 2,1, , ,.....
2 4
Example 2: Does the following geometric series
converge?
6, -9, 13.5, …..
Example 3: Write this decimal as the ratio of two
integers using a geometric series approach.
.5656565656565…..
Example 4: Write this decimal as the ratio of two
relatively prime integers using a series approach.
.4789898989898989…..
1. Define factorial in your own words, and give 2
examples.
2. Write the mathematical notation for a factorial.
3. What is zero factorial
11.7 Applications of Sequences and Series
1. Rework Example 1 on page 597 Assume the first
impact moves the nail 15 millimeters into the wood
and the second impact moves the nail 12 millimeters
into the wood. Model the nail movement as an arith
sequence and then as geo sequence
2. One of the common applications of partial sums of
series is compound interest. Explain in words how
compound interest works (GES)? How is it different
from simple interest?
Example 2: Suppose you invest $500 at the
beginning of the year and the bank pays 4% interest
compounded annually. How much money will you
have at the end of a year. Make a table of the
amounts of money you will have at the end of each of
the first six years. How many years will it take to
triple your investment?
Example3: Suppose the money you invested in
problem one had interest compounded quarterly
instead of annually. Make a table of how much
money you would have at the end of each of the first
6 years. How long would it take to triple your
money?
11.9 and 11.10 Binomial Series and the
Binomial Formula
1. Write out the tenth eleventh and twelfth rows in
Pascal's triangle.
2. Rewrite the rows in combination format.
3. Using the triangles you made write out (a+b)10
4. Using the triangles you made write out (a-b)11
5. Using the triangles you made, write out (2x-3b)5
6. Using the triangles you made, write out (3x2- 12 y3)7
Example 1: Find the term in (a-b)15 which contains b6
Example 2: Find the seventh term of (a+b)11
Example 3: Find the 11th term of (r4-3s)14
7. Copy the blue box in Example 4 p. 622
Example 5: Expand (2r3-s)6 as a binomial series.