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12.2 Geometric Sequences and Series

12.2 Geometric Sequences and Series.

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12.2 Geometric Sequences and Series

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  1. 12.2 Geometric Sequences and Series 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…”.

  2. 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 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” Geometric Sequences

  3. 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…”. • 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 Geometric Formulas(you will get these on the quarterly, quiz and exam)

  4. 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…”. 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. We prefer “times 1/5” to “divided by 5” for terminology because we have “times r” Example 1: Identify the common ratio

  5. 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…”. • 20th term for Which means What is the value of r? Example 2: Find the nth term given each of the following: (use the calculator!)

  6. 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…”. • 15th term for What is the value of r? Example 2: Find the nth term given each of the following: (use the calculator!)

  7. 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…”. • 25th term for What is the value of r? Example 2: Find the nth term given each of the following: (use the calculator!)

  8. 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…”. • First 10 terms of What is the value of r? Example 3: Find the nth partial sum given each of the following: (use the calculator!!)

  9. 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…”. • First 6 terms of What is the value of r? Example 3: Find the nth partial sum given each of the following: (use the calculator!!)

  10. 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…”. • First 8 terms of What is the value of r? Example 3: Find the nth partial sum given each of the following: (use the calculator!!)

  11. 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…”. • Infinite sums can ONLY be found if What is the value of r? Example 4: Find the sum of the infinite geometric series (12.3)

  12. 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…”. • Infinite sums can ONLY be found if What is the value of r? Example 4: Find the sum of the infinite geometric series (12.3)

  13. 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…”. • Infinite sums can ONLY be found if What is the value of r? Example 4: Find the sum of the infinite geometric series (12.3)

  14. 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…”. • Determine the common ratio of the geometric series: • What is the sum of the first ten terms of the series: • The sum of the first 9 terms of a geometric series with is . What is the first term? • 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? Summary

  15. 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…”. • Determine the common ratio of the geometric series: • What is the sum of the first ten terms of the series: • The sum of the first 9 terms of a geometric series with is . What is the first term? • 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? Summary

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