Transcript 11.3-11.4: Geometric Sequences & Series
11.3: Geometric Sequences & Series
n
th Term of an Geometric Sequence:
a
n
= a 1
r
(n – 1)
n
th Term of an Geometric Sequence:
a
n
= a 1
r
(n – 1)
Ex. 2 Determine the following: a) Find the eighth term of a geometric sequence for which
a
1 = -3
and
r = -2
.
n
th Term of an Geometric Sequence:
a
n
= a 1
r
(n – 1)
Ex. 2 Determine the following: a) Find the eighth term of a geometric sequence for which
a
1
a
n
= a 1
r
(n – 1) = -3
and
r = -2
.
n
th Term of an Geometric Sequence:
a
n
= a 1
r
(n – 1)
Ex. 2 Determine the following: a) Find the eighth term of a geometric sequence for which
a
1
a
n
= a 1
r
(n – 1)
a
8 = a 1
r
( 8 – 1) = -3
and
r = -2
.
n
th Term of an Geometric Sequence:
a
n
= a 1
r
(n – 1)
Ex. 2 Determine the following: a) Find the eighth term of a geometric sequence for which
a
1
a
n
= a 1
r
(n – 1)
a a
8 8 = a 1
r
(8 – 1) = (-3)
r
(8 – 1) = -3
and
r = -2
.
n
th Term of an Geometric Sequence:
a
n
= a 1
r
(n – 1)
Ex. 2 Determine the following: a) Find the eighth term of a geometric sequence for which
a
1
a
n
= a 1
r
(n – 1)
a a
8 8 = a 1
r
(8 – 1) = (-3)r (8 – 1)
a
8 = (-3) (-2) (8 – 1) = -3
and
r = -2
.
n
th Term of an Geometric Sequence:
a
n
= a 1
r
(n – 1)
Ex. 2 Determine the following: a) Find the eighth term of a geometric sequence for which
a
1
a
n
= a 1
r
(n – 1)
a a
8 8 = a 1
r
(8 – 1) = (-3)r (8 – 1)
a
8 = (-3)(-2) ( 8 – 1 )
a
8 = (-3)(-2 7 ) = -3
and
r = -2
.
n
th Term of an Geometric Sequence:
a
n
= a 1
r
(n – 1)
Ex. 2 Determine the following: a) Find the eighth term of a geometric sequence for which
a a a a a
n
8 8 8 8 = a 1
r
(n – 1) = a 1
r
(8 – 1) = (-3)r (8 – 1)
a
= (-3)(-2) (8 – 1) = (-3)( -2 7 ) 1 = -3
and
r = -2
.
a
8 = (-3)( -128 )
n
th Term of an Geometric Sequence:
a n
= a 1
r
(n – 1)
Ex. 2 Determine the following: a) Find the eighth term of a geometric sequence for which
a a a a a n
8 8 8 8 = a 1
r
(n – 1) = a 1
r
(8 – 1) = (-3)r (8 – 1) = (-3)(-2) (8 – 1) = (-3)(-2 7 )
a
1 = -3
and
r = -2
.
a
8
a
8 = (-3)(-128) = 384
b) Write an equation for the
n
th term of the geometric sequence 3, 12, 48, 192, …
b) Write an equation for the
n
th term of the geometric sequence 3, 12, 48, 192, … 12 3
b) Write an equation for the
n
th term of the geometric sequence 3, 12, 48, 192, …
·
4
b) Write an equation for the
n
th term of the geometric sequence 3, 12, 48, 192, …
r =
4
b) Write an equation for the
n
th term of the geometric sequence 3, 12, 48, 192, …
a
n
= a 1
r
(n – 1)
r =
4
b) Write an equation for the
n
th term of the geometric sequence 3 , 12, 48, 192, …
a
n
a
n
= a 1
r
(n – 1) = (3)
r
(n – 1)
r =
4
b) Write an equation for the
n
th term of the geometric sequence 3, 12, 48, 192, …
r =
4
a
n
a
n
= a 1
r
(n – 1) = (3) (4) (n – 1)
b) Write an equation for the
n
th term of the geometric sequence 3, 12, 48, 192, …
a
n
a
n
= a 1
r
(n – 1) = (3)4
n
– 1
r =
4
b) Write an equation for the
n
th term of the geometric sequence 3, 12, 48, 192, …
a
n
a
n
= a 1
r
(n – 1) = (3)4
n
– 1
r =
4 c) Find the tenth term of a geometric sequence for which
a
4 = 108
and
r = 3
.
b) Write an equation for the
n
th term of the geometric sequence 3, 12, 48, 192, …
a
n
a
n
= a 1
r
(n – 1) = (3)4
n
– 1
r =
4 c) Find the tenth term of a geometric sequence for which
a
4 = 108
and
r = 3
.
1. Find
a
1
b) Write an equation for the
n
th term of the geometric sequence 3, 12, 48, 192, …
a
n
a
n
= a 1
r
(n – 1) = (3)4
n
– 1
r =
4 c) Find the tenth term of a geometric sequence for which
a
4
1. Find
a
1
:
a
n
= 108 = a 1
r
(n – 1)
and
r = 3
.
b) Write an equation for the
n
th term of the geometric sequence 3, 12, 48, 192, …
a
n
a
n
= a 1
r
(n – 1) = (3)4
n
– 1
r =
4 c) Find the tenth term of a geometric sequence for which
a
4
1. Find
a
1
:
a
4 = 108 = a 1
r
( 4 – 1)
and
r = 3
.
b) Write an equation for the
n
th term of the geometric sequence 3, 12, 48, 192, …
a
n
a
n
= a 1
r
(n – 1) = (3)4
n
– 1
r =
4 c) Find the tenth term of a geometric sequence for which
a
4
1. Find
a
1
:
a
4 = 108 = a 1
r
(4 – 1) 108 = a 1
r
(4 – 1)
and
r = 3
.
b) Write an equation for the
n
th term of the geometric sequence 3, 12, 48, 192, …
a
n
a
n
= a 1
r
(n – 1) = (3)4
n
– 1
r =
4 c) Find the tenth term of a geometric sequence for which
a
4
1. Find
a
1
:
a
4 = 108 = a 1
r
(4 – 1) 108 = a 1
r
(4 – 1) 108 = a 1 3 (4 – 1)
and
r = 3
.
b) Write an equation for the
n
th term of the geometric sequence 3, 12, 48, 192, …
r =
4
a
n
a
n
= a 1
r
(n – 1) = (3)4
n
– 1
c) Find the tenth term of a geometric sequence for which
a
4
1. Find
a
1
:
a
4 = 108 = a 1
r
(4 – 1) 108 = a 1
r
(4 – 1) 108 = a 1 3 ( 4 – 1 )
and
r = 3
.
108 = a 1 3 3
b) Write an equation for the
n
th term of the geometric sequence 3, 12, 48, 192, …
a
n
a
n
= a 1
r
(n = (3)4
n
– 1) – 1
r =
4 c) Find the tenth term of a geometric sequence for which 1. Find
a
1
:
a
4
a
4 = 108 = a 1
r
(4 – 1) 108 = a 1
r
(4 – 1) 108 = a 1 3 (4 – 1)
and
r = 3
.
108 = a 1 3 3 108 = 27
a
1
b) Write an equation for the
n
th term of the geometric sequence 3, 12, 48, 192, …
r =
4
a n a n
= a 1
r
(n = (3)4
n
– 1) – 1
c) Find the tenth term of a geometric sequence for which 1. Find
a
1
:
a
4
a
4 = 108 = a 1
r
(4 – 1) 108 = a 1
r
(4 – 1) 108 = a 1 3 (4 – 1)
and
r = 3
.
108 = a 1 3 3 108 = 27a 1 4 = a 1
c)Find the tenth term of a geometric sequence for which
a
4
1. Find
a
1
:
a
4 = 108 = a 1
r
(4 – 1) 108 = a 1
r
(4 – 1) 108 = a 1 3 (4 – 1)
and
r = 3
.
108 = a 1 3 3 108 = 27a 1 4 = a 1
2. Now use
a
1 = 4
and
r = 3
to find
a
10
c)Find the tenth term of a geometric sequence for which
a
4
1. Find
a
1
:
a
4 = 108 = a 1
r
(4 – 1) 108 = a 1
r
(4 – 1) 108 = a 1 3 (4 – 1)
and
r = 3
.
108 = a 1 3 3 108 = 27a 1 4 = a 1
2. Now use
a
1 = 4
and
r = 3
a
n
= a 1
r
(n – 1)
to find
a
10
c)Find the tenth term of a geometric sequence for which
a
4
1. Find
a
1
:
a
4 = 108 = a 1
r
(4 – 1) 108 = a 1
r
(4 – 1) 108 = a 1 3 (4 – 1)
and
r = 3
.
108 = a 1 3 3 108 = 27a 1 4 = a 1
2. Now use
a
1 = 4
and
r = 3
a
n
= a 1
r
(n – 1)
a
10 = a 1
r
( 10 – 1)
to find
a
10
c)Find the tenth term of a geometric sequence for which 1. Find
a
1
:
a
4
a
4 = 108 = a 1
r
(4 – 1) 108 = a 1
r
(4 – 1) 108 = a 1 3 (4 – 1) 108 = a 1 3 3
and
r = 3
.
108 = 27a 1 4 = a 1
2. Now use
a
1 = 4
a
n
a
10 = a 1
r
(n – 1) = a 1
r
(10 – 1)
a
10
and
r = 3
= (4)
r
(10 – 1)
to find
a
10
c)Find the tenth term of a geometric sequence for which
a
4 = 108
and
r = 3
.
1. Find
a
1
:
a
4 = a 1
r
(4 – 1) 108 = a 1
r
(4 – 1) 108 = a 1 3 (4 – 1) 108 = a 1 3 3 108 = 27a 1 4 = a 1
2. Now use
a
1 = 4
and
r = 3
a n
= a 1
r
(n – 1)
a a
10 10 = a 1
r
(10 – 1) = (4)r (10 – 1)
a
10 = (4) (3) (10 – 1)
to find
a
10
c)Find the tenth term of a geometric sequence for which
a
4 = 108
and
r = 3
.
1. Find
a
1
:
a
4 = a 1
r
(4 – 1) 108 = a 1
r
(4 – 1) 108 = a 1 3 (4 – 1) 108 = a 1 3 3 108 = 27a 1 4 = a 1
2. Now use
a
1 = 4
and
r = 3
a a
10
a a n
10 10 = a 1
r
(n – 1) = a 1
r
(10 – 1) = (4)r (10 – 1) = (4)(3) ( 10 – 1 )
to find
a
10
a
10 = (4)(3 9 )
c)Find the tenth term of a geometric sequence for which
a
4 = 108
and
r = 3
.
1. Find
a
1
:
a
4 = a 1
r
(4 – 1) 108 = a 1
r
(4 – 1) 108 = a 1 3 (4 – 1) 108 = a 1 3 3 108 = 27a 1 4 = a 1
2. Now use
a
1 = 4
and
r = 3
a a
10
a a n
10 10 = a 1
r
(n – 1) = a 1
r
(10 – 1) = (4)r (10 – 1) = (4)(3) (10 – 1)
to find
a
10
a
10 = (4)( 3 9 )
a
10 = (4)( 19,683 )
c)Find the tenth term of a geometric sequence for which
a
4 = 108
and
r = 3
.
1. Find
a
1
:
a
4 = a 1
r
(4 – 1) 108 = a 1
r
(4 – 1) 108 = a 1 3 (4 – 1) 108 = a 1 3 3 108 = 27a 1 4 = a 1
2. Now use
a
1 = 4
and
r = 3
a n a
10 = a 1
r
(n – 1) = a 1
r
(10 – 1)
a
10 = (4)r (10 – 1)
a
10
to find
= (4)(3) (10 – 1)
a
10
a
10 = (4)(3 9 )
a
10
a
10 = (4)(19,683) = 78,732
Sum of a Geometric Series
The sum
S
n
of the first
n
terms of a geometric series is given by the following:
S
n = a 1 (1 – r 1 – r
n
)
Sum of a Geometric Series
The sum
S
n
of the first
n
terms of a geometric series is given by the following:
S
n = a 1 (1 – r 1 – r
n
)
Ex. 3 Find the sum of the first 15 terms of the geometric sequence in which
r = 2
.
a
1 = 2
and
Sum of a Geometric Series
The sum
S
n
of the first
n
terms of a geometric series is given by the following:
S
n = a 1 (1 – r 1 – r
n
)
Ex. 3 Find the sum of the first 15 terms of the geometric sequence in which
r = 2
.
a
1 = 2
and
S
n = a 1 (1 – r 1 – r
n
)
Sum of a Geometric Series
The sum
S
n
of the first
n
terms of a geometric series is given by the following:
S
n = a 1 (1 – r 1 – r
n
)
Ex. 3 Find the sum of the first
r = 2
.
15 terms of the geometric sequence in which
a
1 = 2
and
S
n
S
15 = a 1 (1 – r 1 – r
n
) = a 1 (1 – r
15
) 1 – r
Sum of a Geometric Series
The sum
S
n
of the first
n
terms of a geometric series is given by the following:
S
n = a 1 (1 – r 1 – r
n
)
Ex. 3 Find the sum of the first 15 terms of the geometric sequence in which
a
1 = 2
and
r = 2
.
S
n = a 1 (1 – r 1 – r
n
)
S S
15 15 = a 1 (1 – r
15
) 1 – r = 2 (1 – r
15
) 1 – r
Sum of a Geometric Series
The sum
S
n
of the first
n
terms of a geometric series is given by the following:
S
n = a 1 (1 – r 1 – r
n
)
Ex. 3 Find the sum of the first 15 terms of the geometric sequence in which
a
1 = 2
and
r = 2
.
S
n = a 1 (1 – r 1 – r
n
)
S
15 = a 1 (1 – r 15 ) 1 – r
S S
15 15 = 2(1 – r 15 ) 1 – r = 2(1 – 2 15 ) 1 – 2
Sum of a Geometric Series
The sum given by the following:
S
n
S
n
of the first
= a 1 (1 – r 1 – r
n
)
n
terms of a geometric series is Ex. 3 Find the sum of the first 15 terms of the geometric sequence in which
a
1 = 2
and
r = 2
.
S
n = a 1 (1 – r 1 – r
n
)
S S
15 15 = a 1 (1 – r 15 ) 1 – r = 2(1 – r 15 )
S
15 1 – r = 2(1 – 2 15 )
S
15 1 – 2 = 2(1 – 2 15 ) = 2(1 – 1 – 2 -1 32,768 )
Sum of a Geometric Series
The sum given by the following:
S
n
S
n
of the first
= a 1 (1 – r 1 – r
n
)
n
terms of a geometric series is Ex. 3 Find the sum of the first 15 terms of the geometric sequence in which
a
1 = 2
and
r = 2
.
S
n = a 1 (1 – r 1 – r
n
)
S S
15 15 = a 1 (1 – r 15 ) 1 – r = 2(1 – r 15 )
S
15 1 – r = 2(1 – 2 15 )
S
15 1 – 2 = 2(1 – 2 15 ) = 2(1 – 32,768) = -65,534 1 – 2 -1 -1
Sum of a Geometric Series
The sum given by the following:
S
n
S
n
of the first
= a 1 (1 – r 1 – r
n
)
n
terms of a geometric series is Ex. 3 Find the sum of the first 15 terms of the geometric sequence in which
a
1 = 2
and
r = 2
.
S
n = a 1 (1 – r 1 – r
n
)
S S
15 15 = a 1 (1 – r 15 ) 1 – r = 2(1 – r 15 )
S
15 1 – r = 2(1 – 2 15 )
S
15 1 – 2 = 2(1 – 2 15 ) = 2(1 – 32,768) = -65,534 1 – 2 -1 -1 = 65,534