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83 Geometric Sequencesand Series
Objectives-Students will recognize write and find the nth terms of geometric sequences-Students will find the nth partial sums of geometric sequences-Students will find sums of infinite geometric series-Students will use geometric sequences to model and solve real-life problems
Definition of Geometric Sequence
bull A sequence is geometric if the ratios of consecutive terms are the same
bull a1 a2 a3 a4 hellip an hellip is geometric if there is a number r such that
bull r is the common ratio of the geometric sequence
32 4
1 2 3
0aa a
r ra a a
Ex 1) Determine whether the sequence is geometric If it is find the common ratio
a) 6 18 30 42 hellip
b) 1 -frac12 frac14 -⅛ hellip
The nth Term of a Geometric Sequence
The nth term of a geometric sequence has the form
an = a1 r n ndash 1
where r is the common ratio
Ex 2) Write the first 5 terms of the geometric sequence Find the common ratio and write the nth term of the sequence as a function of n
a1 = 64 and ak+1 = frac12 ak
Ex 3) Find the nth term of the geometric sequence Use the table feature of your calculator to verify your answer
a1 =4 r = frac12 n = 10
Ex 4) Find the nth term of the geometric sequence Use the table feature of your calculator to verify your answer
a2 = -18 a5 =23 n = 6
The Sum of an Infinite Geometric Sequence
If r lt 1
If r gt 1 the series does not have a sum
The Sum of a Finite Geometric Sequence
1
1
1
n
n
rS a
r
1
1
aS
r
Ex 5) Find the sum Use a calculator to verify9
1
1
2n
n
Ex 6) Find the sum Use a calculator to verify
5
0
300(106)n
n
Ex 7) Use summation notation to write the sum
5 + 15 + 45 + hellip + 3645
Ex 8) Find the sum of the infinite geometric series if possible If not possible explain why
0
1
2n
n
Ex 9) Find the sum of the infinite geometric series if possible If not possible explain why
1
17
32
n
n
Ex 10) Find the rational number representation of the repeating decimal
a)
b) 0318
036
Definition of Geometric Sequence
bull A sequence is geometric if the ratios of consecutive terms are the same
bull a1 a2 a3 a4 hellip an hellip is geometric if there is a number r such that
bull r is the common ratio of the geometric sequence
32 4
1 2 3
0aa a
r ra a a
Ex 1) Determine whether the sequence is geometric If it is find the common ratio
a) 6 18 30 42 hellip
b) 1 -frac12 frac14 -⅛ hellip
The nth Term of a Geometric Sequence
The nth term of a geometric sequence has the form
an = a1 r n ndash 1
where r is the common ratio
Ex 2) Write the first 5 terms of the geometric sequence Find the common ratio and write the nth term of the sequence as a function of n
a1 = 64 and ak+1 = frac12 ak
Ex 3) Find the nth term of the geometric sequence Use the table feature of your calculator to verify your answer
a1 =4 r = frac12 n = 10
Ex 4) Find the nth term of the geometric sequence Use the table feature of your calculator to verify your answer
a2 = -18 a5 =23 n = 6
The Sum of an Infinite Geometric Sequence
If r lt 1
If r gt 1 the series does not have a sum
The Sum of a Finite Geometric Sequence
1
1
1
n
n
rS a
r
1
1
aS
r
Ex 5) Find the sum Use a calculator to verify9
1
1
2n
n
Ex 6) Find the sum Use a calculator to verify
5
0
300(106)n
n
Ex 7) Use summation notation to write the sum
5 + 15 + 45 + hellip + 3645
Ex 8) Find the sum of the infinite geometric series if possible If not possible explain why
0
1
2n
n
Ex 9) Find the sum of the infinite geometric series if possible If not possible explain why
1
17
32
n
n
Ex 10) Find the rational number representation of the repeating decimal
a)
b) 0318
036
Ex 1) Determine whether the sequence is geometric If it is find the common ratio
a) 6 18 30 42 hellip
b) 1 -frac12 frac14 -⅛ hellip
The nth Term of a Geometric Sequence
The nth term of a geometric sequence has the form
an = a1 r n ndash 1
where r is the common ratio
Ex 2) Write the first 5 terms of the geometric sequence Find the common ratio and write the nth term of the sequence as a function of n
a1 = 64 and ak+1 = frac12 ak
Ex 3) Find the nth term of the geometric sequence Use the table feature of your calculator to verify your answer
a1 =4 r = frac12 n = 10
Ex 4) Find the nth term of the geometric sequence Use the table feature of your calculator to verify your answer
a2 = -18 a5 =23 n = 6
The Sum of an Infinite Geometric Sequence
If r lt 1
If r gt 1 the series does not have a sum
The Sum of a Finite Geometric Sequence
1
1
1
n
n
rS a
r
1
1
aS
r
Ex 5) Find the sum Use a calculator to verify9
1
1
2n
n
Ex 6) Find the sum Use a calculator to verify
5
0
300(106)n
n
Ex 7) Use summation notation to write the sum
5 + 15 + 45 + hellip + 3645
Ex 8) Find the sum of the infinite geometric series if possible If not possible explain why
0
1
2n
n
Ex 9) Find the sum of the infinite geometric series if possible If not possible explain why
1
17
32
n
n
Ex 10) Find the rational number representation of the repeating decimal
a)
b) 0318
036
The nth Term of a Geometric Sequence
The nth term of a geometric sequence has the form
an = a1 r n ndash 1
where r is the common ratio
Ex 2) Write the first 5 terms of the geometric sequence Find the common ratio and write the nth term of the sequence as a function of n
a1 = 64 and ak+1 = frac12 ak
Ex 3) Find the nth term of the geometric sequence Use the table feature of your calculator to verify your answer
a1 =4 r = frac12 n = 10
Ex 4) Find the nth term of the geometric sequence Use the table feature of your calculator to verify your answer
a2 = -18 a5 =23 n = 6
The Sum of an Infinite Geometric Sequence
If r lt 1
If r gt 1 the series does not have a sum
The Sum of a Finite Geometric Sequence
1
1
1
n
n
rS a
r
1
1
aS
r
Ex 5) Find the sum Use a calculator to verify9
1
1
2n
n
Ex 6) Find the sum Use a calculator to verify
5
0
300(106)n
n
Ex 7) Use summation notation to write the sum
5 + 15 + 45 + hellip + 3645
Ex 8) Find the sum of the infinite geometric series if possible If not possible explain why
0
1
2n
n
Ex 9) Find the sum of the infinite geometric series if possible If not possible explain why
1
17
32
n
n
Ex 10) Find the rational number representation of the repeating decimal
a)
b) 0318
036
Ex 2) Write the first 5 terms of the geometric sequence Find the common ratio and write the nth term of the sequence as a function of n
a1 = 64 and ak+1 = frac12 ak
Ex 3) Find the nth term of the geometric sequence Use the table feature of your calculator to verify your answer
a1 =4 r = frac12 n = 10
Ex 4) Find the nth term of the geometric sequence Use the table feature of your calculator to verify your answer
a2 = -18 a5 =23 n = 6
The Sum of an Infinite Geometric Sequence
If r lt 1
If r gt 1 the series does not have a sum
The Sum of a Finite Geometric Sequence
1
1
1
n
n
rS a
r
1
1
aS
r
Ex 5) Find the sum Use a calculator to verify9
1
1
2n
n
Ex 6) Find the sum Use a calculator to verify
5
0
300(106)n
n
Ex 7) Use summation notation to write the sum
5 + 15 + 45 + hellip + 3645
Ex 8) Find the sum of the infinite geometric series if possible If not possible explain why
0
1
2n
n
Ex 9) Find the sum of the infinite geometric series if possible If not possible explain why
1
17
32
n
n
Ex 10) Find the rational number representation of the repeating decimal
a)
b) 0318
036
Ex 3) Find the nth term of the geometric sequence Use the table feature of your calculator to verify your answer
a1 =4 r = frac12 n = 10
Ex 4) Find the nth term of the geometric sequence Use the table feature of your calculator to verify your answer
a2 = -18 a5 =23 n = 6
The Sum of an Infinite Geometric Sequence
If r lt 1
If r gt 1 the series does not have a sum
The Sum of a Finite Geometric Sequence
1
1
1
n
n
rS a
r
1
1
aS
r
Ex 5) Find the sum Use a calculator to verify9
1
1
2n
n
Ex 6) Find the sum Use a calculator to verify
5
0
300(106)n
n
Ex 7) Use summation notation to write the sum
5 + 15 + 45 + hellip + 3645
Ex 8) Find the sum of the infinite geometric series if possible If not possible explain why
0
1
2n
n
Ex 9) Find the sum of the infinite geometric series if possible If not possible explain why
1
17
32
n
n
Ex 10) Find the rational number representation of the repeating decimal
a)
b) 0318
036
Ex 4) Find the nth term of the geometric sequence Use the table feature of your calculator to verify your answer
a2 = -18 a5 =23 n = 6
The Sum of an Infinite Geometric Sequence
If r lt 1
If r gt 1 the series does not have a sum
The Sum of a Finite Geometric Sequence
1
1
1
n
n
rS a
r
1
1
aS
r
Ex 5) Find the sum Use a calculator to verify9
1
1
2n
n
Ex 6) Find the sum Use a calculator to verify
5
0
300(106)n
n
Ex 7) Use summation notation to write the sum
5 + 15 + 45 + hellip + 3645
Ex 8) Find the sum of the infinite geometric series if possible If not possible explain why
0
1
2n
n
Ex 9) Find the sum of the infinite geometric series if possible If not possible explain why
1
17
32
n
n
Ex 10) Find the rational number representation of the repeating decimal
a)
b) 0318
036
The Sum of an Infinite Geometric Sequence
If r lt 1
If r gt 1 the series does not have a sum
The Sum of a Finite Geometric Sequence
1
1
1
n
n
rS a
r
1
1
aS
r
Ex 5) Find the sum Use a calculator to verify9
1
1
2n
n
Ex 6) Find the sum Use a calculator to verify
5
0
300(106)n
n
Ex 7) Use summation notation to write the sum
5 + 15 + 45 + hellip + 3645
Ex 8) Find the sum of the infinite geometric series if possible If not possible explain why
0
1
2n
n
Ex 9) Find the sum of the infinite geometric series if possible If not possible explain why
1
17
32
n
n
Ex 10) Find the rational number representation of the repeating decimal
a)
b) 0318
036
Ex 5) Find the sum Use a calculator to verify9
1
1
2n
n
Ex 6) Find the sum Use a calculator to verify
5
0
300(106)n
n
Ex 7) Use summation notation to write the sum
5 + 15 + 45 + hellip + 3645
Ex 8) Find the sum of the infinite geometric series if possible If not possible explain why
0
1
2n
n
Ex 9) Find the sum of the infinite geometric series if possible If not possible explain why
1
17
32
n
n
Ex 10) Find the rational number representation of the repeating decimal
a)
b) 0318
036
Ex 6) Find the sum Use a calculator to verify
5
0
300(106)n
n
Ex 7) Use summation notation to write the sum
5 + 15 + 45 + hellip + 3645
Ex 8) Find the sum of the infinite geometric series if possible If not possible explain why
0
1
2n
n
Ex 9) Find the sum of the infinite geometric series if possible If not possible explain why
1
17
32
n
n
Ex 10) Find the rational number representation of the repeating decimal
a)
b) 0318
036
Ex 7) Use summation notation to write the sum
5 + 15 + 45 + hellip + 3645
Ex 8) Find the sum of the infinite geometric series if possible If not possible explain why
0
1
2n
n
Ex 9) Find the sum of the infinite geometric series if possible If not possible explain why
1
17
32
n
n
Ex 10) Find the rational number representation of the repeating decimal
a)
b) 0318
036
Ex 8) Find the sum of the infinite geometric series if possible If not possible explain why
0
1
2n
n
Ex 9) Find the sum of the infinite geometric series if possible If not possible explain why
1
17
32
n
n
Ex 10) Find the rational number representation of the repeating decimal
a)
b) 0318
036
Ex 9) Find the sum of the infinite geometric series if possible If not possible explain why
1
17
32
n
n
Ex 10) Find the rational number representation of the repeating decimal
a)
b) 0318
036
Ex 10) Find the rational number representation of the repeating decimal
a)
b) 0318
036
