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12.2 Geometric Sequences and Series. - PowerPoint PPT Presentation
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By the end of the section students will be able to determine whether a sequence is geometric, find the nth term in a geometric sequence, find the partial sum of a geometric series and find the sum of an infinite geometric series, as evidenced by completion of I have…, who has…”.
*12.2 Geometric Sequences and Series
Geometric Sequences
*What is the next term in the sequence?
*These are all GEOMETRIC sequences because they MULTIPLY or DIVIDE by the same amount. (however, we say multiply by instead of divide by 2, even though they are the same)
*This amount is called the “common ratio”
By the end of the section students will be able to determine whether a sequence is geometric, find the nth term in a geometric sequence, find the partial sum of a geometric series and find the sum of an infinite geom. series, as evidenced by completion of I have…, who has…”.
Geometric Formulas(you will get these on the quarterly, quiz and exam)
* means the nth term* is the common ratio * is the sum of the first n terms.* is the infinite sum (12.3 formula)Nth term
Partial Sum
Infinite Sum
By the end of the section students will be able to determine whether a sequence is geometric, find the nth term in a geometric sequence, find the partial sum of a geometric series and find the sum of an infinite geom. series, as evidenced by completion of I have…, who has…”.
Example 1: Identify the common ratio
Because the ratio is common to each consecutive term, each pair of elements will give you the same ratio.We can also do this “in our head” if the numbers are simple enough.
A.
We prefer “times 1/5” to “divided by 5” for terminology because we have “times r”
B.
C.
By the end of the section students will be able to determine whether a sequence is geometric, find the nth term in a geometric sequence, find the partial sum of a geometric series and find the sum of an infinite geom. series, as evidenced by completion of I have…, who has…”.
Example 2: Find the nth term given each of the following: (use the calculator!)
A. 20th term for
Which means
By the end of the section students will be able to determine whether a sequence is geometric, find the nth term in a geometric sequence, find the partial sum of a geometric series and find the sum of an infinite geom. series, as evidenced by completion of I have…, who has…”.
What is the value of r?
Example 2: Find the nth term given each of the following: (use the calculator!)
B. 15th term for
By the end of the section students will be able to determine whether a sequence is geometric, find the nth term in a geometric sequence, find the partial sum of a geometric series and find the sum of an infinite geom. series, as evidenced by completion of I have…, who has…”.
What is the value of r?
Example 2: Find the nth term given each of the following: (use the calculator!)
C. 25th term for
By the end of the section students will be able to determine whether a sequence is geometric, find the nth term in a geometric sequence, find the partial sum of a geometric series and find the sum of an infinite geom. series, as evidenced by completion of I have…, who has…”.
What is the value of r?
Example 3: Find the nth partial sum given each of the following: (use the calculator!!) A. First 10 terms of
By the end of the section students will be able to determine whether a sequence is geometric, find the nth term in a geometric sequence, find the partial sum of a geometric series and find the sum of an infinite geom. series, as evidenced by completion of I have…, who has…”.
What is the value of r?
Example 3: Find the nth partial sum given each of the following: (use the calculator!!) B. First 6 terms of
By the end of the section students will be able to determine whether a sequence is geometric, find the nth term in a geometric sequence, find the partial sum of a geometric series and find the sum of an infinite geom. series, as evidenced by completion of I have…, who has…”.
What is the value of r?
Example 3: Find the nth partial sum given each of the following: (use the calculator!!) C. First 8 terms of
By the end of the section students will be able to determine whether a sequence is geometric, find the nth term in a geometric sequence, find the partial sum of a geometric series and find the sum of an infinite geom. series, as evidenced by completion of I have…, who has…”.
What is the value of r?
Example 4: Find the sum of the infinite geometric series (12.3)
*Infinite sums can ONLY be found if
A.
By the end of the section students will be able to determine whether a sequence is geometric, find the nth term in a geometric sequence, find the partial sum of a geometric series and find the sum of an infinite geom. series, as evidenced by completion of I have…, who has…”.
What is the value
of r?
Example 4: Find the sum of the infinite geometric series (12.3)
*Infinite sums can ONLY be found if
B.
By the end of the section students will be able to determine whether a sequence is geometric, find the nth term in a geometric sequence, find the partial sum of a geometric series and find the sum of an infinite geom. series, as evidenced by completion of I have…, who has…”.
What is the value
of r?
Example 4: Find the sum of the infinite geometric series (12.3)
*Infinite sums can ONLY be found if
C.
By the end of the section students will be able to determine whether a sequence is geometric, find the nth term in a geometric sequence, find the partial sum of a geometric series and find the sum of an infinite geom. series, as evidenced by completion of I have…, who has…”.
What is the value
of r?
Summary
1. Determine the common ratio of the geometric series:
2. What is the sum of the first ten terms of the series: 3. The sum of the first 9 terms of a geometric series
with is . What is the first term?4. Ms. Munch spins VERY fast on her rolly chair, at a
maximum of 120 revolutions per minute. Suppose she gets tired and decides to stop, she slows down by 5/8ths rotation per minute. How many complete revolutions will she make once she decides to stop spinning?
By the end of the section students will be able to determine whether a sequence is geometric, find the nth term in a geometric sequence, find the partial sum of a geometric series and find the sum of an infinite geom. series, as evidenced by completion of I have…, who has…”.
Summary1. Determine the common ratio of the geometric series:
2. What is the sum of the first ten terms of the series:
3. The sum of the first 9 terms of a geometric series with is . What is the first term?
4. Ms. Munch spins VERY fast on her rolly chair, at a maximum of 120 revolutions per minute. Suppose she gets tired and decides to stop, she slows down by 5/8ths rotation per minute. How many complete revolutions will she make once she decides to stop spinning?
By the end of the section students will be able to determine whether a sequence is geometric, find the nth term in a geometric sequence, find the partial sum of a geometric series and find the sum of an infinite geom. series, as evidenced by completion of I have…, who has…”.
