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Geometric Sequen= ces
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A sequence a1, a2 , a3 , … is an geometric sequence if
an =3D= crn, i.e. it is defined by an exponential formula.
Geometric Sequen= ces
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Example A. The sequence of powers of 2
a1=3D 2, a2=3D 4, a3=3D 8, a4=3D 16, = 230;
is an geometric sequence because an =3D 2n.
A sequence a1, a2 , a3 , … is an geometric sequence if
an =3D= crn, i.e. it is defined by an exponential formula.
Geometric Sequen= ces
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Example A. The sequence of powers of 2
a1=3D 2, a2=3D 4, a3=3D 8, a4=3D 16, = 230;
is an geometric sequence because an =3D 2n.
Fac= t: If a1, a2 , a= 3 , …an =3D c*rn is a geometric sequence, then the ratio between any two consecutive terms is r.
A sequence a1, a2 , a3 , … is an geometric sequence if
an =3D= crn, i.e. it is defined by an exponential formula.
Geometric Sequen= ces
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Example A. The sequence of powers of 2
a1=3D 2, a2=3D 4, a3=3D 8, a4=3D 16, = 230;
is an geometric sequence because an =3D 2n.
A sequence a1, a2 , a3 , … is an geometric sequence if
an =3D= crn, i.e. it is defined by an exponential formula.
Geometric Sequen= ces
The converse of this fact= is also true.
Fac= t: If a1, a2 , a= 3 , …an =3D c*rn is a geometric sequence, then the ratio between any two consecutive terms is r.
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Example A. The sequence of powers of 2
a1=3D 2, a2=3D 4, a3=3D 8, a4=3D 16, = 230;
is an geometric sequence because an =3D 2n.
A sequence a1, a2 , a3 , … is an geometric sequence if
an =3D= crn, i.e. it is defined by an exponential formula.
Geometric Sequen= ces
The converse of this fact is also true. Below is the formula that we used for working = with geometric sequences.
Fac= t: If a1, a2 , a= 3 , …an =3D c*rn is a geometric sequence, then the ratio between any two consecutive terms is r.
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Example A. The sequence of powers of 2
a1=3D 2, a2=3D 4, a3=3D 8, a4=3D 16, = 230;
is an geometric sequence because an =3D 2n.
A sequence a1, a2 , a3 , … is an geometric sequence if
an =3D= crn, i.e. it is defined by an exponential formula.
Geometric Sequen= ces
The converse of this fact= is also true.
For example, from the sequence above we see that
16/8 =3D 8/4 =3D 4/2 =3D 2 =3D = ratio r. Below is = the formula that we used for working with geometric sequences. = ;
Fac= t: If a1, a2 , a= 3 , …an =3D c*rn is a geometric sequence, then the ratio between any two consecutive terms is r.
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Example A. The sequence of powers of 2
a1=3D 2, a2=3D 4, a3=3D 8, a4=3D 16, = 230;
is an geometric sequence because an =3D 2n.
A sequence a1, a2 , a3 , … is an geometric sequence if
an =3D= crn, i.e. it is defined by an exponential formula.
Geometric Sequen= ces
Theorem: If a1, a2 , a3 , …an is a sequence such that
an+1 / an =3D= r for all n, then a= 1, a2, a3,&#= 8230; is  an geometric sequence and an =3D a1*rn-1.
The converse of this fact is also true. Below is the formula that we used for working = with geometric sequences.
For example, from the sequence above we see that
16/8 =3D 8/4 =3D 4/2 =3D 2 =3D = ratio r
Fac= t: If a1, a2 , a= 3 , …an =3D c*rn is a geometric sequence, then the ratio between any two consecutive terms is r.
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Example A. The sequence of powers of 2
a1=3D 2, a2=3D 4, a3=3D 8, a4=3D 16, = 230;
is an geometric sequence because an =3D 2n.
A sequence a1, a2 , a3 , … is an geometric sequence if
an =3D= crn, i.e. it is defined by an exponential formula.
Geometric Sequen= ces
Theorem: If a1, a2 , a3 , …an is a sequence such that
an+1 / an =3D= r for all n, then a= 1, a2, a3,&#= 8230; is  an geometric sequence and an =3D a1*rn-1. This is the general formula for geometric sequences.
The converse of this fact is also true. Below is the formula that we used for working = with geometric sequences.
For example, from the sequence above we see that
16/8 =3D 8/4 =3D 4/2 =3D 2 =3D = ratio r
Fac= t: If a1, a2 , a= 3 , …an =3D c*rn is a geometric sequence, then the ratio between any two consecutive terms is r.
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Geometric Sequen= ces
Giv= en the description of a geometric sequence, we use the general formula to find t= he specific for= mula for that sequence. 
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Example A. The sequence 2, 6, 18, 54, … is an geometric sequence because 6/2 =3D 18/6 = ;
Geometric Sequen= ces
Giv= en the description of a geometric sequence, we use the general formula to find t= he specific for= mula for that sequence. 
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Example A. The sequence 2, 6, 18, 54, … is an geometric sequence because 6/2 =3D 18/6 =3D 54/18 = ;
Geometric Sequen= ces
Giv= en the description of a geometric sequence, we use the general formula to find t= he specific for= mula for that sequence. 
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Example A. The sequence 2, 6, 18, 54, … is an geometric sequence because 6/2 =3D 18/6 =3D 54/18 =3D … =3D 3 =3D r.
Geometric Sequen= ces
Giv= en the description of a geometric sequence, we use the general formula to find t= he specific for= mula for that sequence. 
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Example A. The sequence 2, 6, 18, 54, … is an geometric sequence because 6/2 =3D 18/6 =3D 54/18 =3D … =3D 3 =3D r. =
Since a1 = =3D 2, set them in the general formula of the geometric sequenc= es an = =3D a1r n ̵= 1; 1
Geometric Sequen= ces
Giv= en the description of a geometric sequence, we use the general formula to find t= he specific for= mula for that sequence. 
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Example A. The sequence 2, 6, 18, 54, … is an geometric sequence because 6/2 =3D 18/6 =3D 54/18 =3D … =3D 3 =3D r. =
Since a1 = =3D 2, set them in the general formula of the geometric sequenc= es an = =3D a1r n ̵= 1; 1 , we get the specific formula for this sequence  an =3D 2*3(n – 1) 
Geometric Sequen= ces
Giv= en the description of a geometric sequence, we use the general formula to find t= he specific for= mula for that sequence. 
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Example A. The sequence 2, 6, 18, 54, … is an geometric sequence because 6/2 =3D 18/6 =3D 54/18 =3D … =3D 3 =3D r. =
Since a1 = =3D 2, set them in the general formula of the geometric sequenc= es an = =3D a1r n ̵= 1; 1 , we get the specific formula for this sequence  an =3D 2*3(n – 1) 
Geometric Sequen= ces
If a1, a= 2 , a3 , …an is a geometric sequence such that = the terms alternate between positive and negative signs,
then r is negative.
Giv= en the description of a geometric sequence, we use the general formula to find t= he specific for= mula for that sequence. 
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Example A. The sequence 2, 6, 18, 54, … is an geometric sequence because 6/2 =3D 18/6 =3D 54/18 =3D … =3D 3 =3D r. =
Since a1 = =3D 2, set them in the general formula of the geometric sequenc= es an = =3D a1r n ̵= 1; 1 , we get the specific formula for this sequence  an =3D 2*3(n – 1) 
Geometric Sequen= ces
If a1, a= 2 , a3 , …an is a geometric sequence such that = the terms alternate between positive and negative signs,
then r is negative.
Example B. The sequence 2/3, –1, 3/2, –9/4, … is a geometric sequence because
Giv= en the description of a geometric sequence, we use the general formula to find t= he specific for= mula for that sequence. 
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Example A. The sequence 2, 6, 18, 54, … is an geometric sequence because 6/2 =3D 18/6 =3D 54/18 =3D … =3D 3 =3D r. =
Since a1 = =3D 2, set them in the general formula of the geometric sequenc= es an = =3D a1r n ̵= 1; 1 , we get the specific formula for this sequence  an =3D 2*3(n – 1) 
Geometric Sequen= ces
If a1, a= 2 , a3 , …an is a geometric sequence such that = the terms alternate between positive and negative signs,
then r is negative.
Example B. The sequence 2/3, –1, 3/2, –9/4, … is a geometric sequence because
–1 /(2/3)        = ;            &n= bsp;            = ;  
Giv= en the description of a geometric sequence, we use the general formula to find t= he specific for= mula for that sequence. 
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Example A. The sequence 2, 6, 18, 54, … is an geometric sequence because 6/2 =3D 18/6 =3D 54/18 =3D … =3D 3 =3D r. =
Since a1 = =3D 2, set them in the general formula of the geometric sequenc= es an = =3D a1r n ̵= 1; 1 , we get the specific formula for this sequence  an =3D 2*3(n – 1) 
Geometric Sequen= ces
If a1, a= 2 , a3 , …an is a geometric sequence such that = the terms alternate between positive and negative signs,
then r is negative.
Example B. The sequence 2/3, –1, 3/2, –9/4, … is a geometric sequence because
–1 /(= 2/3) =3D (3/2) / (–1)
Giv= en the description of a geometric sequence, we use the general formula to find t= he specific for= mula for that sequence. 
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Example A. The sequence 2, 6, 18, 54, … is an geometric sequence because 6/2 =3D 18/6 =3D 54/18 =3D … =3D 3 =3D r. =
Since a1 = =3D 2, set them in the general formula of the geometric sequenc= es an = =3D a1r n ̵= 1; 1 , we get the specific formula for this sequence  an =3D 2*3(n – 1) 
Geometric Sequen= ces
If a1, a= 2 , a3 , …an is a geometric sequence such that = the terms alternate between positive and negative signs,
then r is negative.
Example B. The sequence 2/3, –1, 3/2, –9/4, … is a geometric sequence because
–1 /(= 2/3) =3D (3/2) / (–1) =3D (–9/4) /(3/2) =3D … =3D –3/2 =3D r.<= /span>= 3;
Giv= en the description of a geometric sequence, we use the general formula to find t= he specific for= mula for that sequence. 
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Example A. The sequence 2, 6, 18, 54, … is an geometric sequence because 6/2 =3D 18/6 =3D 54/18 =3D … =3D 3 =3D r. =
Since a1 = =3D 2, set them in the general formula of the geometric sequenc= es an = =3D a1r n ̵= 1; 1 , we get the specific formula for this sequence  an =3D 2*3(n – 1) 
Geometric Sequen= ces
If a1, a= 2 , a3 , …an is a geometric sequence such that = the terms alternate between positive and negative signs,
then r is negative.
Example B. The sequence 2/3, –1, 3/2, –9/4, … is a geometric sequence because
–1 /(= 2/3) =3D (3/2) / (–1) =3D (–9/4) /(3/2) =3D … =3D –3/2 =3D r. = ;
Since a1 = =3D 2/3, hence the specific formula is
=
        = ;            &n= bsp;            = ;    an =3D    (     )n–1
2
3
2
–3
Giv= en the description of a geometric sequence, we use the general formula to find t= he specific for= mula for that sequence. 
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Geometric Sequen= ces
To = use the geometric general formula to find the specific formula, we need the first term a1 and the ratio r.
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Example C. Given that a1, a<= span lang=3DEN-US style=3D'mso-fareast-font-family:\5B8B\4F53;mso-hansi-font-fa= mily: Arial;font-size:89%;color:#003300;position:relative;top:.37em;mso-text-rai= se: -25%;mso-fareast-language:ZH-CN'>2 , a3 , …is= a geometric sequence with r =3D –2 and a5  = =3D 12.
Geometric Sequen= ces
To = use the geometric general formula to find the specific formula, we need the first term a1 and the ratio r.
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Example C. Given = that a1, a2 , a<= span lang=3DEN-US style=3D'mso-fareast-font-family:\5B8B\4F53;mso-hansi-font-fa= mily: Arial;font-size:89%;color:#003300;position:relative;top:.37em;mso-text-rai= se: -25%;mso-fareast-language:ZH-CN'>3 , …is a geometric sequence with r =3D –2 and a5  =3D 12.
=
a. find a<= span lang=3DEN-US style=3D'mso-fareast-font-family:\5B8B\4F53;mso-hansi-font-fa= mily: Arial;font-size:89%;color:#003300;position:relative;top:.37em;mso-text-rai= se: -25%;mso-fareast-language:ZH-CN'>1.   
Geometric Sequen= ces
To = use the geometric general formula to find the specific formula, we need the first term a1 and the ratio r.
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Geometric Sequen= ces
To = use the geometric general formula to find the specific formula, we need the first term a1 and the ratio r.
Example C. Given = that a1, a2 , a<= span lang=3DEN-US style=3D'mso-fareast-font-family:\5B8B\4F53;mso-hansi-font-fa= mily: Arial;font-size:89%;color:#003300;position:relative;top:.37em;mso-text-rai= se: -25%;mso-fareast-language:ZH-CN'>3 , …is a geometric sequence with r =3D –2 and a5  =3D 12.
=
a. find a<= span lang=3DEN-US style=3D'mso-fareast-font-family:\5B8B\4F53;mso-hansi-font-fa= mily: Arial;font-size:89%;color:#003300;position:relative;top:.37em;mso-text-rai= se: -25%;mso-fareast-language:ZH-CN'>1.    
Set n =3D = 5 in the gen= eral geometric formula an =3D a1r n ̵= 1; 1,
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Geometric Sequen= ces
To = use the geometric general formula to find the specific formula, we need the first term a1 and the ratio r.
Example C. Given = that a1, a2 , a<= span lang=3DEN-US style=3D'mso-fareast-font-family:\5B8B\4F53;mso-hansi-font-fa= mily: Arial;font-size:89%;color:#003300;position:relative;top:.37em;mso-text-rai= se: -25%;mso-fareast-language:ZH-CN'>3 , …is a geometric sequence with r =3D –2 and a5  =3D 12.
=
a. find a<= span lang=3DEN-US style=3D'mso-fareast-font-family:\5B8B\4F53;mso-hansi-font-fa= mily: Arial;font-size:89%;color:#003300;position:relative;top:.37em;mso-text-rai= se: -25%;mso-fareast-language:ZH-CN'>1.    
Set n =3D = 5 in the gen= eral geometric formula an =3D a1r n ̵= 1; 1,
we get
a5 =3D a1(–2)(5 – 1) =3D 12
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Geometric Sequen= ces
To = use the geometric general formula to find the specific formula, we need the first term a1 and the ratio r.
Example C. Given = that a1, a2 , a<= span lang=3DEN-US style=3D'mso-fareast-font-family:\5B8B\4F53;mso-hansi-font-fa= mily: Arial;font-size:89%;color:#003300;position:relative;top:.37em;mso-text-rai= se: -25%;mso-fareast-language:ZH-CN'>3 , …is a geometric sequence with r =3D –2 and a5  =3D 12.
=
a. find a<= span lang=3DEN-US style=3D'mso-fareast-font-family:\5B8B\4F53;mso-hansi-font-fa= mily: Arial;font-size:89%;color:#003300;position:relative;top:.37em;mso-text-rai= se: -25%;mso-fareast-language:ZH-CN'>1.    
Set n =3D = 5 in the gen= eral geometric formula an =3D a1r n ̵= 1; 1,
we get
a5 =3D a1(–2)(5 – 1) =3D 12
        = ;     a1(–2)<= /span>4 = =3D 12
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Geometric Sequen= ces
To = use the geometric general formula to find the specific formula, we need the first term a1 and the ratio r.
Example C. Given = that a1, a2 , a<= span lang=3DEN-US style=3D'mso-fareast-font-family:\5B8B\4F53;mso-hansi-font-fa= mily: Arial;font-size:89%;color:#003300;position:relative;top:.37em;mso-text-rai= se: -25%;mso-fareast-language:ZH-CN'>3 , …is a geometric sequence with r =3D –2 and a5  =3D 12.
=
a. find a<= span lang=3DEN-US style=3D'mso-fareast-font-family:\5B8B\4F53;mso-hansi-font-fa= mily: Arial;font-size:89%;color:#003300;position:relative;top:.37em;mso-text-rai= se: -25%;mso-fareast-language:ZH-CN'>1.    
Set n =3D = 5 in the gen= eral geometric formula an =3D a1r n ̵= 1; 1,
we get
a5 =3D a1(–2)(5 – 1) =3D 12
        = ;     a1(–2)<= /span>4 = =3D 12
        = ;        16a1 =3D 12
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Geometric Sequen= ces
To = use the geometric general formula to find the specific formula, we need the first term a1 and the ratio r.
Example C. Given = that a1, a2 , a<= span lang=3DEN-US style=3D'mso-fareast-font-family:\5B8B\4F53;mso-hansi-font-fa= mily: Arial;font-size:89%;color:#003300;position:relative;top:.37em;mso-text-rai= se: -25%;mso-fareast-language:ZH-CN'>3 , …is a geometric sequence with r =3D –2 and a5  =3D 12.
=
a. find a<= span lang=3DEN-US style=3D'mso-fareast-font-family:\5B8B\4F53;mso-hansi-font-fa= mily: Arial;font-size:89%;color:#003300;position:relative;top:.37em;mso-text-rai= se: -25%;mso-fareast-language:ZH-CN'>1.    
Set n =3D = 5 in the gen= eral geometric formula an =3D a1r n ̵= 1; 1,
we get
a5 =3D a1(–2)(5 – 1) =3D 12
        = ;     a1(–2)<= /span>4 = =3D 12
        = ;        16a1 =3D 12
        = ;           a1 =3D 12/16 =3D ¾
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Geometric Sequen= ces
To = use the geometric general formula to find the specific formula, we need the first term a1 and the ratio r.
b. find the specific equation.        = ;   
Example C. Given = that a1, a2 , a<= span lang=3DEN-US style=3D'mso-fareast-font-family:\5B8B\4F53;mso-hansi-font-fa= mily: Arial;font-size:89%;color:#003300;position:relative;top:.37em;mso-text-rai= se: -25%;mso-fareast-language:ZH-CN'>3 , …is a geometric sequence with r =3D –2 and a5  =3D 12.
=
a. find a<= span lang=3DEN-US style=3D'mso-fareast-font-family:\5B8B\4F53;mso-hansi-font-fa= mily: Arial;font-size:89%;color:#003300;position:relative;top:.37em;mso-text-rai= se: -25%;mso-fareast-language:ZH-CN'>1.    
Set n =3D = 5 in the gen= eral geometric formula an =3D a1r n ̵= 1; 1,
we get
a5 =3D a1(–2)(5 – 1) =3D 12
        = ;     a1(–2)<= /span>4 = =3D 12
        = ;        16a1 =3D 12
        = ;           a1 =3D 12/16 =3D ¾
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Geometric Sequen= ces
To = use the geometric general formula to find the specific formula, we need the first term a1 and the ratio r.
b. find the specific equation.        = ;   
Set a1 = =3D ¾ and r =3D –2 into the general formula an = =3D a1rn ̵= 1; 1 ,
Example C. Given = that a1, a2 , a<= span lang=3DEN-US style=3D'mso-fareast-font-family:\5B8B\4F53;mso-hansi-font-fa= mily: Arial;font-size:89%;color:#003300;position:relative;top:.37em;mso-text-rai= se: -25%;mso-fareast-language:ZH-CN'>3 , …is a geometric sequence with r =3D –2 and a5  =3D 12.
=
a. find a<= span lang=3DEN-US style=3D'mso-fareast-font-family:\5B8B\4F53;mso-hansi-font-fa= mily: Arial;font-size:89%;color:#003300;position:relative;top:.37em;mso-text-rai= se: -25%;mso-fareast-language:ZH-CN'>1.    
Set n =3D = 5 in the gen= eral geometric formula an =3D a1r n ̵= 1; 1,
we get
a5 =3D a1(–2)(5 – 1) =3D 12
        = ;     a1(–2)<= /span>4 = =3D 12
        = ;        16a1 =3D 12
        = ;           a1 =3D 12/16 =3D ¾
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3
4
an=3D      (–2= )n–1
Geometric Sequen= ces
To = use the geometric general formula to find the specific formula, we need the first term a1 and the ratio r.
b. find the specific equation.        = ;   
Set a1 = =3D ¾ and r =3D –2 into the general formula an = =3D a1rn ̵= 1; 1 ,
we get the specif= ic formula of this sequence
Example C. Given = that a1, a2 , a<= span lang=3DEN-US style=3D'mso-fareast-font-family:\5B8B\4F53;mso-hansi-font-fa= mily: Arial;font-size:89%;color:#003300;position:relative;top:.37em;mso-text-rai= se: -25%;mso-fareast-language:ZH-CN'>3 , …is a geometric sequence with r =3D –2 and a5  =3D 12.
=
a. find a<= span lang=3DEN-US style=3D'mso-fareast-font-family:\5B8B\4F53;mso-hansi-font-fa= mily: Arial;font-size:89%;color:#003300;position:relative;top:.37em;mso-text-rai= se: -25%;mso-fareast-language:ZH-CN'>1.    
Set n =3D = 5 in the gen= eral geometric formula an =3D a1r n ̵= 1; 1,
we get
a5 =3D a1(–2)(5 – 1) =3D 12
        = ;     a1(–2)<= /span>4 = =3D 12
        = ;        16a1 =3D 12
        = ;           a1 =3D 12/16 =3D ¾
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C.  Find a9.
Geometric Sequen= ces
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C.  Find a9.
Geometric Sequen= ces
3
4
Since an=3D      (–2= )n–1,
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set n =3D <= /span>9, we get
C.  Find a9.
3
4
a9=3D       (–2)9–1 =
Geometric Sequen= ces
3
4
Since an=3D      (–2= )n–1,
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set n =3D <= /span>9, we get
C.  Find a9.
3
4
a9=3D       (–2)9–1
a9 =3D      (–2= )8
3
4
Geometric Sequen= ces
3
4
Since an=3D      (–2= )n–1,
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set n =3D <= /span>9, we get
C.  Find a9.
3
4
a9=3D       (–2)9–1
a9 =3D      (–2= )8 =3D       (25= 6) =3D 192 
3
4
Geometric Sequen= ces
3
4
Since an=3D      (–2= )n–1,
3
4
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set n =3D <= /span>9, we get
C.  Find a9.
3
4
a9=3D       (–2)9–1
a9 =3D      (–2= )8 =3D       (25= 6) =3D 192 
3
4
Geometric Sequen= ces
3
4
Since an=3D      (–2= )n–1,