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Algebra 2 Interactive Chalkboard
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GLENCOE DIVISION
Glencoe/McGraw-Hill
8787 Orion Place
Columbus, Ohio 43240
Lesson 11-1Arithmetic Sequences
Lesson 11-2Arithmetic Series
Lesson 11-3Geometric Sequences
Lesson 11-4Geometric Series
Lesson 11-5Infinite Geometric Series
Lesson 11-6Recursion and Special Sequences
Lesson 11-7The Binomial Theorem
Lesson 11-8Proof and Mathematical Induction
Example 1 Find the Next Terms
Example 2 Find a Particular Term
Example 3 Write an Equation for the nth Term
Example 4 Find Arithmetic Means
Find the next four terms of the arithmetic sequence
–8, –6, –4, ….
Find the common difference d by subtracting 2
consecutive terms.
Now add 2 to the third term of the sequence and
then continue adding 2 until the next four terms
are found.
–4
–2
+2
0
+2
2
4
+2
Answer: The next four terms are –2, 0, 2 and 4.
+2
Find the next four terms of the arithmetic sequence
5, 3, 1, ….
Answer: The next four terms are –1, –3, –5 and –7.
Construction The table below shows typical costs for
a construction company to rent a crane for one, two,
three, or four months. Assuming that the arithmetic
sequence continues, how much would it cost to rent
the crane for 24 months?
Months
Cost($)
1
75,000
2
3
90,000
105,000
4
120,000
Explore
Plan
Months
Cost($)
1
2
75,000
90,000
3
4
105,000
120,000
Since the difference between any two
successive costs is $15,000, the costs form an
arithmetic sequence with common difference
15,000.
You can use the formula for the nth term of an
arithmetic sequence with
and
to find
the cost for
24 months.
Solve
Formula for the nth term
Simplify.
Answer: It would cost $420,000 to rent for 24 months.
Examine You can find the term of the sequence by adding
15,000. From Example 2 on page 579 of your
textbook, you know the cost to rent the crane for
12 months is $120,000. So, a12 through a24 are
240,000, 255,000, 270,000, 285,000, 300,000,
315,000, 330,000, 345,000, 360,000, 375,000,
390,000, 405,000, and 420,000. Therefore,
$420,000 is correct.
Construction The table below shows typical costs for
a construction company to rent a crane for one, two,
three, or four months. Assuming that the arithmetic
sequence continues, how much would it cost to rent
the crane for 8 months?
Months
Cost($)
1
2
3
4
75,000
90,000
105,000
120,000
Answer: It would cost $180,000 to rent for
8 months.
Write an equation for the nth term of the arithmetic
sequence –8, –6, –4, ….
In this sequence,
write an equation.
and
Use the nth formula to
Formula for the nth term
Distributive Property
Simplify.
Answer: An equation is
.
Write an equation for the nth term of the arithmetic
sequence 5, 3, 1, ….
Answer: An equation is
.
Find the three arithmetic means between 21 and 45.
You can use the nth term formula to find the common
difference. In the sequence 21, ___, ___, ___, 45, ...,
and
Formula for the nth term
Subtract 21 from each side.
Divide each side by 4.
Now use the value of d to find the three arithmetic means.
21
27
+6
33
+6
39
+6
Answer: The arithmetic means are 27, 33, and 39.
Check
Find the three arithmetic means between 13 and 25.
Answer: The arithmetic means are 16, 19, and 22.
Example 1 Find the Sum of an Arithmetic Series
Example 2 Find the First Term
Example 3 Find the First Three Terms
Example 4 Evaluate a Sum in Sigma Notation
Find the sum of the first 20 even numbers, beginning
with 2.
The series is
Since you can see that
and
you can use either sum formula
for this series.
Method 1
Sum formula
Simplify.
Multiply.
Method 2
Sum formula
Simplify.
Multiply.
Answer: The sum of the first 20 even numbers
is 420.
Find the sum of the first 15 numbers, beginning with 1.
Answer: The sum of the first 15 numbers
is 120.
Radio A radio station is giving away money every day
in the month of September for a total of $124,000.
They plan to increase the amount of money given away
by $100 each day. How much should they give away on
the first day of September, rounded to the
nearest cent?
You know the values of n, Sn, and d. Use the sum formula
that contains d.
Sum formula
Simplify.
Distributive Property
Subtract 43,500.
Divide by 30.
Answer: They should give away $2683.33 the
first day.
Games A television game show gives contestants a
chance to win a total of $1,000,000 by answering 16
consecutive questions correctly. If the value of each
question is increased by $5,000, how much is the first
question worth?
Answer: The first question is worth $25,000.
Find the first four terms of an arithmetic series in
which
Step 1
Since you know
and
use
Step 2
Find d.
Step 3
Use d to determine
Answer: The first four terms are 14, 17, 20, 23.
Find the first three terms of an arithmetic series in
which
and
Answer: The first three terms are 11, 16, 21.
Evaluate
Method 1 Find the terms by replacing k with 3, 4, …, 10.
Then add.
Method 2 Since the sum is an arithmetic series, use the
formula
There are 8 terms,
Answer: The sum of the series is 112.
Evaluate
Answer: 108
Example 1 Find the Next Term
Example 2 Find a Particular Term
Example 3 Write an Equation for the nth Term
Example 4 Find a Term Given the Fourth Term and
the Ratio
Example 5 Find Geometric Means
Multiple-Choice Test Item
Find the missing term in the geometric sequence 324,
108, 36, 12, ___.
A 972
B 4
C 0
D –12
Read the Test Item
Since
the common ratio of
the sequence has
Solve the Test Item
To find the missing term, multiply the last given term
by
Answer: B
Multiple-Choice Test Item
Find the missing term in the geometric sequence 100,
50, 25, ___.
A 200
Answer: C
B 0
C 12.5
D –12.5
Find the sixth term of a geometric sequence for which
and
Formula for the nth term
Multiply.
Answer: The sixth term is 96.
Find the fifth term of a geometric sequence for which
and
Answer: The fifth term is 96.
Write an equation for the nth term of the geometric
sequence 5, 10, 20, 40, ….
Formula for the nth term
Answer: An equation is
Write an equation for the nth term of the geometric
sequence 2, 6, 18, 54, ….
Answer: An equation is
.
Find the seventh term of a geometric sequence for
which
and
First find the value of
Formula for the nth term
Divide by 4.
Now find a7.
Formula for the nth term
Use a calculator.
Answer: The seventh term is 1536.
Find the sixth term of a geometric sequence for
which
and
Answer: The sixth term is 243.
Find three geometric means between 3.12 and 49.92.
Use the nth term formula to find the value of r. In the
sequence 3.12, ___, ___, ___, 49.92, a1 is 3.12 and a5
is 49.92.
Formula for the nth term
Divide by 3.12.
Take the fourth root of each side.
There are two possible common ratios, so there are two
possible sets of geometric means. Use each value of r
to find three geometric means.
Answer: The geometric means are 6.24, 12.48, and
24.96, or –6.24, 12.48, and –24.96.
Find three geometric means between 12 and 0.75.
Answer: The geometric means are 6, 3, and 1.5, or –6, 3,
and –1.5.
Example 1 Find the Sum of the First n Terms
Example 2 Evaluate a Sum Written in Sigma Notation
Example 3 Use the Alternate Formula for a Sum
Example 4 Find the First Term of a Series
Genealogy How many direct ancestors would a person
have after 8 generations?
Counting two parents, four grandparents, eight greatgrandparents, and so on gives you a geometric series with
Sum formula
Use a calculator.
Answer: Going back 8 generations, a person
would have 510 ancestors.
Genealogy How many direct ancestors would a person
have after 7 generations?
Answer: Going back 7 generations, a person
would have 254 ancestors.
Evaluate
Method 1
Find the terms by replacing n with 1, 2, 3, 4, 5, 6, 7, 8, 9,
10, 11 and 12. Then add.
Method 2
Since the sum is a geometric series, you can use the
formula
Sum formula
Simplify.
Answer: The sum of the series is 12,285.
Evaluate
Answer: 200
Find the sum of a geometric series for which
Since you do not know the value of n, use the formula
Alternate sum formula
Simplify.
Answer: The sum of the series is 6666.
Find the sum of a geometric series for which
Answer: The sum of the series is 74.4.
Find a1 in a geometric series for which
and
Sum formula
Subtract.
Divide each side by –255.
Answer: The first term of the series is 3.
Find a1 in a geometric series for which
and
Answer: The first term of the series is 1.
Example 1 Sum of an Infinite Geometric Series
Example 2 Infinite Series in Sigma Notation
Example 3 Write a Repeating Decimal as a Fraction
Find the sum of
, if
it exists.
First, find the value of r to determine if the sum exists.
the sum does not exist.
Answer: The sum does not exist.
Find the sum of
the sum exists.
, if it exists.
Now use the formula for the sum of an infinite
geometric series.
Sum formula
Simplify.
Answer: The sum of the series is 2.
Find the sum of each infinite geometric series, if
it exists.
a.
Answer: no sum
b.
Answer: 2
Evaluate
In this infinite geometric series,
Sum formula
Simplify.
Answer: Thus,
Evaluate
Answer: 3
Write
as a fraction.
Method 1
Write the repeating decimal as a sum.
In this series,
Sum formula
Subtract.
Simplify.
Method 2
Label the given decimal.
Repeating decimal
Multiply each side by 100.
Subtract the second equation
from the third.
Divide each side by 99.
Answer: Thus,
Write
Answer:
as a fraction.
Example 1 Use a Recursive Formula
Example 2 Find and Use a Recursive Formula
Example 3 Iterate a Function
Find the first five terms of the sequence in which
and
Recursive formula
Recursive formula
Answer: The first five terms of the sequence
are 5, 17, 41, 89, 185.
Find the first five terms of the sequence in which
and
Answer: The first five terms of the sequence
are 2, 8, 26, 80, 242.
Biology Dr. Elliott is growing cells in lab dishes. She
starts with 108 cells Monday morning and then
removes 20 of these for her experiment. By Tuesday
the remaining cells have multiplied by 1.5. She again
removes 20. This pattern repeats each day in the week.
Write a recursive formula for the number of cells Dr.
Elliott finds each day before she removes any.
Let cn represent the number of cells at the beginning of the
nth day. She takes 20 out, leaving cn – 20. The number the
next day will be 1.5 times as much. So,
Answer:
Find the number of cells she will find on
Friday morning.
On the first morning, there were 108 cells, so
Recursive formula
Recursive formula
Answer: On the fifth day, there will 303 cells.
Biology Dr. Scott is growing cells in lab dishes. She
starts with 100 cells Monday morning and then
removes 30 of these for her experiment. By Tuesday
the remaining cells have doubled. She again removes
30. This pattern repeats each day in the week.
a. Write a recursive formula for the number of cells Dr.
Scott finds each day before she removes any.
Answer:
b. Find the number of cells she will find on
Saturday morning.
Answer: 1340
Find the first three iterates x1, x2, and x3 of the function
for an initial value of
To find the first iterate x1, find the value of the function
for
Iterate the function.
Simplify.
To find the second iterate x2, substitute x1 for x.
Iterate the function.
Simplify.
Substitute x2 for x to find the third iterate.
Iterate the function.
Simplify.
Answer: Therefore, 5, 14, 41, 122 is an example of a
sequence generated using iteration.
Find the first three iterates x1, x2, and x3 of the function
for an initial value of
Answer: 5, 11, 23
Example 1 Use Pascal’s Triangle
Example 2 Use the Binomial Theorem
Example 3 Factorials
Example 4 Use a Factorial Form of the
Binomial Theorem
Example 5 Find a Particular Term
Expand
Write row 5 of Pascal’s triangle.
1
5
10
10
5
1
Use the patterns of a binomial expansion and the
coefficients to write the expansion of
Answer:
Expand
Answer:
Expand
The expression will have nine terms. Use the sequence
to find the coefficients
for the first five terms. Use symmetry to find the
remaining coefficients.
Answer:
Expand
Answer:
Evaluate
1
1
Answer:
Evaluate
Answer: 420
Expand
Binomial Theorem,
factorial form
Let
Simplify.
Answer:
Expand
Answer:
Find the fourth term in the expansion of
First, use the Binomial Theorem to write the expression
in sigma notation.
In the fourth term,
Answer:
Simplify.
Find the fifth term in the expansion of
Answer:
Example 1 Summation Formula
Example 2 Divisibility
Example 3 Counterexample
Prove that
Step 1
When,
the left side of the given equation
is 2(1) –1 or 1. The right side is 12 or 1. Thus,
the equation is true for
Step 2
Assume
positive integer k.
Step 3
Show that the given equation is true
for
for a
Add
Add.
Simplify.
Factor.
to each side.
The last expression is the right side of the equation to be
proved, where n has been replaced by
Thus, the
equation is true for
Answer: This proves that
is true for all positive integers n.
Prove that
.
Answer:
Step 1
When,
we have
, which is true.
Thus, the equation is true for
Step 2
Assume that
for a positive integer k.
Step 3
Show that the given equation is true
for
The last expression is the right side of the equation to be
proved, where n has been replaced by
Thus, the
equation is true for
This proves that
is true for all
positive integers n.
Prove that
integers n.
is divisible by 5 for all positive
Step 1
When,
Since 5 is
divisible by 5, the statement is true for
Step 2
Assume that
is divisible by 5 for some
positive integer k. This means that there is a
whole number r such that
Step 3
Show that the statement is true for
Inductive hypothesis
Add 1 to each side.
Multiply each side by 6.
Simplify.
Subtract 1 from each side.
Factor.
Since r is a whole number,
is a whole number.
Therefore
is divisible by 5.
Answer: Thus, the statement is true for
proves that
is divisible by 5.
This
Prove that
integers n.
is divisible by 9 for all positive
Answer:
Step 1
When,
Since 9
is divisible by 9, the statement is true for
Step 2
Assume that
positive integer k.
is divisible by 9 for some
Step 3
Show that this is true for
.
Since r is a whole number,
is a whole number.
Therefore,
is divisible by 9. Thus, the statement
is true for
This proves that
is divisible
by 9.
Find a counterexample for the formula that
is always a prime number for any positive integer n.
Check the first few positive integers.
n
Formula
prime?
1
2
3
4
12 + 1 + 5 or 7
22 + 2 + 5 or 11
32 + 3 + 5 or 17
42 + 4 + 5 or 25
yes
yes
yes
no
Answer: The value
the formula.
is a counterexample for
Find a counterexample for the formula that
is always a prime number for any positive integer n.
Answer: The value
the formula.
is a counterexample for
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