93 geometric sequences
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Geometric Sequences
A sequence a1, a2 , a3 , … is an geometric sequence if an = crn, i.e. it is defined by an exponential formula.
Geometric Sequences
Example A. The sequence of powers of 2a1= 2, a2= 4, a3= 8, a4= 16, … is an geometric sequence because an = 2n.
A sequence a1, a2 , a3 , … is an geometric sequence if an = crn, i.e. it is defined by an exponential formula.
Geometric Sequences
Example A. The sequence of powers of 2a1= 2, a2= 4, a3= 8, a4= 16, … is an geometric sequence because an = 2n.Fact: If a1, a2 , a3 , …an = 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 = crn, i.e. it is defined by an exponential formula.
Geometric Sequences
Example A. The sequence of powers of 2a1= 2, a2= 4, a3= 8, a4= 16, … is an geometric sequence because an = 2n.
A sequence a1, a2 , a3 , … is an geometric sequence if an = crn, i.e. it is defined by an exponential formula.
Geometric Sequences
The converse of this fact is also true.
Fact: If a1, a2 , a3 , …an = c*rn is a geometric sequence, then the ratio between any two consecutive terms is r.
Example A. The sequence of powers of 2a1= 2, a2= 4, a3= 8, a4= 16, … is an geometric sequence because an = 2n.
A sequence a1, a2 , a3 , … is an geometric sequence if an = crn, i.e. it is defined by an exponential formula.
Geometric Sequences
The converse of this fact is also true. Below is the formula that we used for working with geometric sequences.
Fact: If a1, a2 , a3 , …an = c*rn is a geometric sequence, then the ratio between any two consecutive terms is r.
Example A. The sequence of powers of 2a1= 2, a2= 4, a3= 8, a4= 16, … is an geometric sequence because an = 2n.
A sequence a1, a2 , a3 , … is an geometric sequence if an = crn, i.e. it is defined by an exponential formula.
Geometric Sequences
The converse of this fact is also true. For example, from the sequence above we see that 16/8 = 8/4 = 4/2 = 2 = ratio r. Below is the formula that we used for working with geometric sequences.
Fact: If a1, a2 , a3 , …an = c*rn is a geometric sequence, then the ratio between any two consecutive terms is r.
Example A. The sequence of powers of 2a1= 2, a2= 4, a3= 8, a4= 16, … is an geometric sequence because an = 2n.
A sequence a1, a2 , a3 , … is an geometric sequence if an = crn, i.e. it is defined by an exponential formula.
Geometric Sequences
Theorem: If a1, a2 , a3 , …an is a sequence such that an+1 / an = r for all n, then a1, a2, a3,… is an geometric sequence and an = 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 = 8/4 = 4/2 = 2 = ratio r
Fact: If a1, a2 , a3 , …an = c*rn is a geometric sequence, then the ratio between any two consecutive terms is r.
Example A. The sequence of powers of 2a1= 2, a2= 4, a3= 8, a4= 16, … is an geometric sequence because an = 2n.
A sequence a1, a2 , a3 , … is an geometric sequence if an = crn, i.e. it is defined by an exponential formula.
Geometric Sequences
Theorem: If a1, a2 , a3 , …an is a sequence such that an+1 / an = r for all n, then a1, a2, a3,… is an geometric sequence and an = 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 = 8/4 = 4/2 = 2 = ratio r
Fact: If a1, a2 , a3 , …an = c*rn is a geometric sequence, then the ratio between any two consecutive terms is r.
Geometric SequencesGiven the description of a geometric sequence, we use the general formula to find the specific formula for that sequence.
Example A. The sequence 2, 6, 18, 54, … is an geometric sequence because 6/2 = 18/6
Geometric SequencesGiven the description of a geometric sequence, we use the general formula to find the specific formula for that sequence.
Example A. The sequence 2, 6, 18, 54, … is an geometric sequence because 6/2 = 18/6 = 54/18
Geometric SequencesGiven the description of a geometric sequence, we use the general formula to find the specific formula for that sequence.
Example A. The sequence 2, 6, 18, 54, … is an geometric sequence because 6/2 = 18/6 = 54/18 = … = 3 = r.
Geometric SequencesGiven the description of a geometric sequence, we use the general formula to find the specific formula for that sequence.
Example A. The sequence 2, 6, 18, 54, … is an geometric sequence because 6/2 = 18/6 = 54/18 = … = 3 = r.Since a1 = 2, set them in the general formula of the geometric sequences an = a1r n – 1
Geometric SequencesGiven the description of a geometric sequence, we use the general formula to find the specific formula for that sequence.
Example A. The sequence 2, 6, 18, 54, … is an geometric sequence because 6/2 = 18/6 = 54/18 = … = 3 = r.Since a1 = 2, set them in the general formula of the geometric sequences an = a1r n – 1 , we get the specific formula for this sequence an = 2*3(n – 1)
Geometric SequencesGiven the description of a geometric sequence, we use the general formula to find the specific formula for that sequence.
Example A. The sequence 2, 6, 18, 54, … is an geometric sequence because 6/2 = 18/6 = 54/18 = … = 3 = r.Since a1 = 2, set them in the general formula of the geometric sequences an = a1r n – 1 , we get the specific formula for this sequence an = 2*3(n – 1)
Geometric Sequences
If a1, a2 , a3 , …an is a geometric sequence such that the terms alternate between positive and negative signs, then r is negative.
Given the description of a geometric sequence, we use the general formula to find the specific formula for that sequence.
Example A. The sequence 2, 6, 18, 54, … is an geometric sequence because 6/2 = 18/6 = 54/18 = … = 3 = r.Since a1 = 2, set them in the general formula of the geometric sequences an = a1r n – 1 , we get the specific formula for this sequence an = 2*3(n – 1)
Geometric Sequences
If a1, a2 , 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) = (3/2) / (-1)
Given the description of a geometric sequence, we use the general formula to find the specific formula for that sequence.
Example A. The sequence 2, 6, 18, 54, … is an geometric sequence because 6/2 = 18/6 = 54/18 = … = 3 = r.Since a1 = 2, set them in the general formula of the geometric sequences an = a1r n – 1 , we get the specific formula for this sequence an = 2*3(n – 1)
Geometric Sequences
If a1, a2 , 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) = (3/2) / (-1) = (-9/4) /(3/2)
Given the description of a geometric sequence, we use the general formula to find the specific formula for that sequence.
Example A. The sequence 2, 6, 18, 54, … is an geometric sequence because 6/2 = 18/6 = 54/18 = … = 3 = r.Since a1 = 2, set them in the general formula of the geometric sequences an = a1r n – 1 , we get the specific formula for this sequence an = 2*3(n – 1)
Geometric Sequences
If a1, a2 , 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) = (3/2) / (-1) = (-9/4) /(3/2) = … = -3/2 = r.
Given the description of a geometric sequence, we use the general formula to find the specific formula for that sequence.
Example A. The sequence 2, 6, 18, 54, … is an geometric sequence because 6/2 = 18/6 = 54/18 = … = 3 = r.Since a1 = 2, set them in the general formula of the geometric sequences an = a1r n – 1 , we get the specific formula for this sequence an = 2*3(n – 1)
Geometric Sequences
If a1, a2 , 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) = (3/2) / (-1) = (-9/4) /(3/2) = … = -3/2 = r.Since a1 = 2/3, the specific formula is
an = ( )n–1 23 2
–3
Given the description of a geometric sequence, we use the general formula to find the specific formula for that sequence.
Geometric SequencesTo 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 , a3 , …is a geometric sequence with r = -2 and a5 = 12,
Geometric SequencesTo 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 , a3 , …is a geometric sequence with r = -2 and a5 = 12,
a. find a1
Geometric SequencesTo 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 , a3 , …is a geometric sequence with r = -2 and a5 = 12,
a. find a1
By that the general geometric formulaan = a1r n – 1, we get a5 = a1(-2)(5 – 1)
Geometric SequencesTo 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 , a3 , …is a geometric sequence with r = -2 and a5 = 12,
a. find a1
By that the general geometric formulaan = a1r n – 1, we get a5 = a1(-2)(5 – 1) = 12
Geometric SequencesTo 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 , a3 , …is a geometric sequence with r = -2 and a5 = 12,
a. find a1
By that the general geometric formulaan = a1r n – 1, we get a5 = a1(-2)(5 – 1) = 12 a1(-2)4 = 12
Geometric SequencesTo 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 , a3 , …is a geometric sequence with r = -2 and a5 = 12,
a. find a1
By that the general geometric formulaan = a1r n – 1, we get a5 = a1(-2)(5 – 1) = 12 a1(-2)4 = 12 16a1 = 12
Geometric SequencesTo 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 , a3 , …is a geometric sequence with r = -2 and a5 = 12,
a. find a1
By that the general geometric formulaan = a1r n – 1, we get a5 = a1(-2)(5 – 1) = 12 a1(-2)4 = 12 16a1 = 12 a1 = 12/16 = ¾
Geometric SequencesTo 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 , a3 , …is a geometric sequence with r = -2 and a5 = 12,
a. find a1
By that the general geometric formulaan = a1r n – 1, we get a5 = a1(-2)(5 – 1) = 12 a1(-2)4 = 12 16a1 = 12 a1 = 12/16 = ¾
Geometric SequencesTo 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 , a3 , …is a geometric sequence with r = -2 and a5 = 12,
a. find a1
By that the general geometric formulaan = a1r n – 1, we get a5 = a1(-2)(5 – 1) = 12 a1(-2)4 = 12 16a1 = 12 a1 = 12/16 = ¾
Geometric SequencesTo 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 = ¾ and r = -2 into the general formula an = a1rn – 1 ,
Example C. Given that a1, a2 , a3 , …is a geometric sequence with r = -2 and a5 = 12,
a. find a1
By that the general geometric formulaan = a1r n – 1, we get a5 = a1(-2)(5 – 1) = 12 a1(-2)4 = 12 16a1 = 12 a1 = 12/16 = ¾
34
an= (-2)n–1
Geometric SequencesTo 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 = ¾ and r = -2 into the general formula an = a1rn – 1 ,we get the specific formula of this sequence
C. Find a9. Geometric Sequences
C. Find a9. Geometric Sequences
34
Since an= (-2)n–1,
set n = 9, we get
C. Find a9.
34
a9= (-2)9–1
Geometric Sequences
34
Since an= (-2)n–1,
set n = 9, we get
C. Find a9.
34
a9= (-2)9–1
a9 = (-2)834
Geometric Sequences
34
Since an= (-2)n–1,
set n = 9, we get
C. Find a9.
34
a9= (-2)9–1
a9 = (-2)8 = (256) = 192 34
Geometric Sequences
34
Since an= (-2)n–1,
34
set n = 9, we get
C. Find a9.
34
a9= (-2)9–1
a9 = (-2)8 = (256) = 192 34
Geometric Sequences
34
Since an= (-2)n–1,
34
Example D. Given that a1, a2 , a3 , …is an geometric sequence with a3 = -2 and a6 = 54,
set n = 9, we get
C. Find a9.
34
a9= (-2)9–1
a9 = (-2)8 = (256) = 192 34
Geometric Sequences
34
Since an= (-2)n–1,
34
Example D. Given that a1, a2 , a3 , …is an geometric sequence with a3 = -2 and a6 = 54,
a. find r and a1
set n = 9, we get
C. Find a9.
34
a9= (-2)9–1
a9 = (-2)8 = (256) = 192 34
Geometric Sequences
34
Since an= (-2)n–1,
34
Example D. Given that a1, a2 , a3 , …is an geometric sequence with a3 = -2 and a6 = 54,
a. find r and a1
Given that the general geometric formula an = a1rn – 1,
set n = 9, we get
C. Find a9.
34
a9= (-2)9–1
a9 = (-2)8 = (256) = 192 34
Geometric Sequences
34
Since an= (-2)n–1,
34
Example D. Given that a1, a2 , a3 , …is an geometric sequence with a3 = -2 and a6 = 54,
a. find r and a1
Given that the general geometric formula an = a1rn – 1, we have a3 = -2 = a1r3–1
set n = 9, we get
C. Find a9.
34
a9= (-2)9–1
a9 = (-2)8 = (256) = 192 34
Geometric Sequences
34
Since an= (-2)n–1,
34
Example D. Given that a1, a2 , a3 , …is an geometric sequence with a3 = -2 and a6 = 54,
a. find r and a1
Given that the general geometric formula an = a1rn – 1, we have a3 = -2 = a1r3–1 and a6 = 54 = a1r6–1
set n = 9, we get
C. Find a9.
34
a9= (-2)9–1
a9 = (-2)8 = (256) = 192 34
Geometric Sequences
34
Since an= (-2)n–1,
34
Example D. Given that a1, a2 , a3 , …is an geometric sequence with a3 = -2 and a6 = 54,
a. find r and a1
Given that the general geometric formula an = a1rn – 1, we have a3 = -2 = a1r3–1 and a6 = 54 = a1r6–1 -2 = a1r2
set n = 9, we get
C. Find a9.
34
a9= (-2)9–1
a9 = (-2)8 = (256) = 192 34
Geometric Sequences
34
Since an= (-2)n–1,
34
Example D. Given that a1, a2 , a3 , …is an geometric sequence with a3 = -2 and a6 = 54,
a. find r and a1
Given that the general geometric formula an = a1rn – 1, we have a3 = -2 = a1r3–1 and a6 = 54 = a1r6–1 -2 = a1r2 54 = a1r5
set n = 9, we get
C. Find a9.
34
a9= (-2)9–1
a9 = (-2)8 = (256) = 192 34
Geometric Sequences
34
Since an= (-2)n–1,
34
Example D. Given that a1, a2 , a3 , …is an geometric sequence with a3 = -2 and a6 = 54,
a. find r and a1
Given that the general geometric formula an = a1rn – 1, we have a3 = -2 = a1r3–1 and a6 = 54 = a1r6–1 -2 = a1r2 54 = a1r5 Divide these equations:
54-2
=a1r5
a1r2
Geometric Sequences
54-2
=a1r5
a1r2-27
Geometric Sequences
54-2
=a1r5
a1r2-27
Geometric Sequences
54-2
=a1r5
a1r2-27 3 = 5-2
Geometric Sequences
54-2
=a1r5
a1r2-27 3 = 5-2
-27 = r3
Geometric Sequences
54-2
=a1r5
a1r2-27 3 = 5-2
-27 = r3
-3 = r
Geometric Sequences
54-2
=a1r5
a1r2-27 3 = 5-2
-27 = r3
-3 = rPut r = -3 into the equation -2 = a1r2
Geometric Sequences
54-2
=a1r5
a1r2-27 3 = 5-2
-27 = r3
-3 = rPut r = -3 into the equation -2 = a1r2
Hence -2 = a1(-3)2
Geometric Sequences
54-2
=a1r5
a1r2-27 3 = 5-2
-27 = r3
-3 = rPut r = -3 into the equation -2 = a1r2
Hence -2 = a1(-3)2
-2 = a19
Geometric Sequences
54-2
=a1r5
a1r2-27 3 = 5-2
-27 = r3
-3 = rPut r = -3 into the equation -2 = a1r2
Hence -2 = a1(-3)2
-2 = a19 -2/9 = a1
Geometric Sequences
54-2
=a1r5
a1r2-27 3 = 5-2
-27 = r3
-3 = rPut r = -3 into the equation -2 = a1r2
Hence -2 = a1(-3)2
-2 = a19 -2/9 = a1
Geometric Sequences
b. Find the specific formula and a2
54-2
=a1r5
a1r2-27 3 = 5-2
-27 = r3
-3 = rPut r = -3 into the equation -2 = a1r2
Hence -2 = a1(-3)2
-2 = a19 -2/9 = a1
Geometric Sequences
b. Find the specific formula and a2
Use the general geometric formula an = a1rn – 1, set a1 = -2/9, and r = -3
54-2
=a1r5
a1r2-27 3 = 5-2
-27 = r3
-3 = rPut r = -3 into the equation -2 = a1r2
Hence -2 = a1(-3)2
-2 = a19 -2/9 = a1
Geometric Sequences
b. Find the specific formula and a2
Use the general geometric formula an = a1rn – 1, set a1 = -2/9, and r = -3 we have the specific formula
-2 9
an = (-3)n–1
54-2
=a1r5
a1r2-27 3 = 5-2
-27 = r3
-3 = rPut r = -3 into the equation -2 = a1r2
Hence -2 = a1(-3)2
-2 = a19 -2/9 = a1
Geometric Sequences
b. Find the specific formula and a2
Use the general geometric formula an = a1rn – 1, set a1 = -2/9, and r = -3 we have the specific formula
-2 9
an = (-3)n–1
-2 9
(-3) 2–1
To find a2, set n = 2, we get
a2 =
54-2
=a1r5
a1r2-27 3 = 5-2
-27 = r3
-3 = rPut r = -3 into the equation -2 = a1r2
Hence -2 = a1(-3)2
-2 = a19 -2/9 = a1
Geometric Sequences
b. Find the specific formula and a2
Use the general geometric formula an = a1rn – 1, set a1 = -2/9, and r = -3 we have the specific formula
-2 9
an = (-3)n–1
-2 9
(-3) 2–1
To find a2, set n = 2, we get
-2 9a2 = =
(-3)
54-2
=a1r5
a1r2-27 3 = 5-2
-27 = r3
-3 = rPut r = -3 into the equation -2 = a1r2
Hence -2 = a1(-3)2
-2 = a19 -2/9 = a1
Geometric Sequences
b. Find the specific formula and a2
Use the general geometric formula an = a1rn – 1, set a1 = -2/9, and r = -3 we have the specific formula
-2 9
an = (-3)n–1
-2 9
(-3) 2–1
To find a2, set n = 2, we get
-2 9a2 = 3 =
(-3)
54-2
=a1r5
a1r2-27 3 = 5-2
-27 = r3
-3 = rPut r = -3 into the equation -2 = a1r2
Hence -2 = a1(-3)2
-2 = a19 -2/9 = a1
Geometric Sequences
b. Find the specific formula and a2
Use the general geometric formula an = a1rn – 1, set a1 = -2/9, and r = -3 we have the specific formula
-2 9
an = (-3)n–1
-2 9
(-3) 2–1
To find a2, set n = 2, we get
-2 9a2 = 3
2 3 =
(-3) =
Geometric SequencesSum of geometric sequences
We observe the algebraic patterns:(1 – r)(1 + r) = 1 – r2
Geometric SequencesSum of geometric sequences
We observe the algebraic patterns:(1 – r)(1 + r) = 1 – r2
(1 – r)(1 + r + r2) = 1 – r3
Geometric SequencesSum of geometric sequences
We observe the algebraic patterns:(1 – r)(1 + r) = 1 – r2
(1 – r)(1 + r + r2) = 1 – r3
(1 – r)(1 + r + r2 + r3) = 1 – r4
Geometric SequencesSum of geometric sequences
We observe the algebraic patterns:(1 – r)(1 + r) = 1 – r2
(1 – r)(1 + r + r2) = 1 – r3
(1 – r)(1 + r + r2 + r3) = 1 – r4
(1 – r)(1 + r + r2 + r3 + r4) = 1 – r5
Geometric SequencesSum of geometric sequences
We observe the algebraic patterns:(1 – r)(1 + r) = 1 – r2
(1 – r)(1 + r + r2) = 1 – r3
(1 – r)(1 + r + r2 + r3) = 1 – r4
(1 – r)(1 + r + r2 + r3 + r4) = 1 – r5
...…(1 – r)(1 + r + r2 … + rn-1) = 1 – rn
Geometric SequencesSum of geometric sequences
We observe the algebraic patterns:(1 – r)(1 + r) = 1 – r2
(1 – r)(1 + r + r2) = 1 – r3
(1 – r)(1 + r + r2 + r3) = 1 – r4
(1 – r)(1 + r + r2 + r3 + r4) = 1 – r5
...…(1 – r)(1 + r + r2 … + rn-1) = 1 – rn
Hence 1 + r + r2 + … + rn-1 = 1 – rn
1 – r
Geometric SequencesSum of geometric sequences
We observe the algebraic patterns:(1 – r)(1 + r) = 1 – r2
(1 – r)(1 + r + r2) = 1 – r3
(1 – r)(1 + r + r2 + r3) = 1 – r4
(1 – r)(1 + r + r2 + r3 + r4) = 1 – r5
...…(1 – r)(1 + r + r2 … + rn-1) = 1 – rn
Hence 1 + r + r2 + … + rn-1 = 1 – rn
1 – r
Geometric SequencesSum of geometric sequences
Therefore a1 + a1r + a1r2 + … +a1rn-1
We observe the algebraic patterns:(1 – r)(1 + r) = 1 – r2
(1 – r)(1 + r + r2) = 1 – r3
(1 – r)(1 + r + r2 + r3) = 1 – r4
(1 – r)(1 + r + r2 + r3 + r4) = 1 – r5
...…(1 – r)(1 + r + r2 … + rn-1) = 1 – rn
Hence 1 + r + r2 + … + rn-1 = 1 – rn
1 – r
Geometric SequencesSum of geometric sequences
Therefore a1 + a1r + a1r2 + … +a1rn-1
n terms
We observe the algebraic patterns:(1 – r)(1 + r) = 1 – r2
(1 – r)(1 + r + r2) = 1 – r3
(1 – r)(1 + r + r2 + r3) = 1 – r4
(1 – r)(1 + r + r2 + r3 + r4) = 1 – r5
...…(1 – r)(1 + r + r2 … + rn-1) = 1 – rn
Hence 1 + r + r2 + … + rn-1 = 1 – rn
1 – r
Geometric SequencesSum of geometric sequences
Therefore a1 + a1r + a1r2 + … +a1rn-1
= a1(1 + r + r2 + … + r n-1) n terms
We observe the algebraic patterns:(1 – r)(1 + r) = 1 – r2
(1 – r)(1 + r + r2) = 1 – r3
(1 – r)(1 + r + r2 + r3) = 1 – r4
(1 – r)(1 + r + r2 + r3 + r4) = 1 – r5
...…(1 – r)(1 + r + r2 … + rn-1) = 1 – rn
Hence 1 + r + r2 + … + rn-1 = 1 – rn
1 – r
Geometric SequencesSum of geometric sequences
a11 – rn
1 – r
Therefore a1 + a1r + a1r2 + … +a1rn-1
= a1(1 + r + r2 + … + r n-1) = n terms
= a11 – rn
1 – r
Formula for the Sum Geometric Sequences
a1 + a1r + a1r2 + … +a1rn-1
Geometric Sequences
= a11 – rn
1 – r
Formula for the Sum Geometric Sequences
a1 + a1r + a1r2 + … +a1rn-1
Geometric Sequences
Example E. Find the geometric sum :2/3 + (-1) + 3/2 + … + (-81/16)
= a11 – rn
1 – r
Formula for the Sum Geometric Sequences
a1 + a1r + a1r2 + … +a1rn-1
Geometric Sequences
Example E. Find the geometric sum :2/3 + (-1) + 3/2 + … + (-81/16)
We have a1 = 2/3 and r = -3/2,
= a11 – rn
1 – r
Formula for the Sum Geometric Sequences
a1 + a1r + a1r2 + … +a1rn-1
Geometric Sequences
Example E. Find the geometric sum :2/3 + (-1) + 3/2 + … + (-81/16)
We have a1 = 2/3 and r = -3/2, and an = -81/16.
= a11 – rn
1 – r
Formula for the Sum Geometric Sequences
a1 + a1r + a1r2 + … +a1rn-1
Geometric Sequences
Example E. Find the geometric sum :2/3 + (-1) + 3/2 + … + (-81/16)
We have a1 = 2/3 and r = -3/2, and an = -81/16. We need the number of terms.
= a11 – rn
1 – r
Formula for the Sum Geometric Sequences
a1 + a1r + a1r2 + … +a1rn-1
Geometric Sequences
Example E. Find the geometric sum :2/3 + (-1) + 3/2 + … + (-81/16)
We have a1 = 2/3 and r = -3/2, and an = -81/16. We need the number of terms. Put a1 and r in the general formula we get the specific formula
23
- 32an= ( ) n-1
= a11 – rn
1 – r
Formula for the Sum Geometric Sequences
a1 + a1r + a1r2 + … +a1rn-1
Geometric Sequences
Example E. Find the geometric sum :2/3 + (-1) + 3/2 + … + (-81/16)
We have a1 = 2/3 and r = -3/2, and an = -81/16. We need the number of terms. Put a1 and r in the general formula we get the specific formula
23
- 32an= ( ) n-1
To find n, set an = -8116
= a11 – rn
1 – r
Formula for the Sum Geometric Sequences
a1 + a1r + a1r2 + … +a1rn-1
Geometric Sequences
Example E. Find the geometric sum :2/3 + (-1) + 3/2 + … + (-81/16)
We have a1 = 2/3 and r = -3/2, and an = -81/16. We need the number of terms. Put a1 and r in the general formula we get the specific formula
23
- 32an= ( ) n-1
To find n, set an = = 23
- 32 ( ) n – 1 -81
16
= a11 – rn
1 – r
Formula for the Sum Geometric Sequences
a1 + a1r + a1r2 + … +a1rn-1
Geometric Sequences
Example E. Find the geometric sum :2/3 + (-1) + 3/2 + … + (-81/16)
We have a1 = 2/3 and r = -3/2, and an = -81/16. We need the number of terms. Put a1 and r in the general formula we get the specific formula
23
- 32an= ( ) n-1
To find n, set an = = 23
- 32 ( ) n – 1 -81
16- 32= ( ) n – 1 -243
32
= a11 – rn
1 – r
Formula for the Sum Geometric Sequences
a1 + a1r + a1r2 + … +a1rn-1
Geometric Sequences
Example E. Find the geometric sum :2/3 + (-1) + 3/2 + … + (-81/16)
We have a1 = 2/3 and r = -3/2, and an = -81/16. We need the number of terms. Put a1 and r in the general formula we get the specific formula
23
- 32an= ( ) n-1
To find n, set an = = 23
- 32 ( ) n – 1 -81
16- 32= ( ) n – 1 -243
32Compare the denominators we see that 32 = 2n – 1.
= a11 – rn
1 – r
Formula for the Sum Geometric Sequences
a1 + a1r + a1r2 + … +a1rn-1
Geometric Sequences
Example E. Find the geometric sum :2/3 + (-1) + 3/2 + … + (-81/16)
We have a1 = 2/3 and r = -3/2, and an = -81/16. We need the number of terms. Put a1 and r in the general formula we get the specific formula
23
- 32an= ( ) n-1
To find n, set an = = 23
- 32 ( ) n – 1 -81
16- 32= ( ) n – 1 -243
32Compare the denominators we see that 32 = 2n – 1.Since 32 = 25 = 2n – 1
= a11 – rn
1 – r
Formula for the Sum Geometric Sequences
a1 + a1r + a1r2 + … +a1rn-1
Geometric Sequences
Example E. Find the geometric sum :2/3 + (-1) + 3/2 + … + (-81/16)
We have a1 = 2/3 and r = -3/2, and an = -81/16. We need the number of terms. Put a1 and r in the general formula we get the specific formula
23
- 32an= ( ) n-1
To find n, set an = = 23
- 32 ( ) n – 1 -81
16- 32= ( ) n – 1 -243
32Compare the denominators we see that 32 = 2n – 1.Since 32 = 25 = 2n – 1
n – 1 = 5
= a11 – rn
1 – r
Formula for the Sum Geometric Sequences
a1 + a1r + a1r2 + … +a1rn-1
Geometric Sequences
Example E. Find the geometric sum :2/3 + (-1) + 3/2 + … + (-81/16)
We have a1 = 2/3 and r = -3/2, and an = -81/16. We need the number of terms. Put a1 and r in the general formula we get the specific formula
23
- 32an= ( ) n-1
To find n, set an = = 23
- 32 ( ) n – 1 -81
16- 32= ( ) n – 1 -243
32Compare the denominators we see that 32 = 2n – 1.Since 32 = 25 = 2n – 1
n – 1 = 5 n = 6
Therefore there are 6 terms in the sum, 2/3 + (-1) + 3/2 + … + (-81/16)
Geometric Sequences
Therefore there are 6 terms in the sum, 2/3 + (-1) + 3/2 + … + (-81/16)
Geometric Sequences
Set a1 = 2/3, r = -3/2 and n = 6 in the formula 1 – rn
1 – rS = a1
Therefore there are 6 terms in the sum, 2/3 + (-1) + 3/2 + … + (-81/16)
S = 231 – (-3/2)6
1 – (-3/2)
Geometric Sequences
Set a1 = 2/3, r = -3/2 and n = 6 in the formula 1 – rn
1 – rS = a1
we get the sum S
Therefore there are 6 terms in the sum, 2/3 + (-1) + 3/2 + … + (-81/16)
S = 231 – (-3/2)6
1 – (-3/2)
= 23
1 – (729/64)1 + (3/2)
Geometric Sequences
Set a1 = 2/3, r = -3/2 and n = 6 in the formula 1 – rn
1 – rS = a1
we get the sum S
Therefore there are 6 terms in the sum, 2/3 + (-1) + 3/2 + … + (-81/16)
S = 231 – (-3/2)6
1 – (-3/2)
= 23
1 – (729/64)1 + (3/2)
= 23
-665/645/2
Geometric Sequences
Set a1 = 2/3, r = -3/2 and n = 6 in the formula 1 – rn
1 – rS = a1
we get the sum S
Therefore there are 6 terms in the sum, 2/3 + (-1) + 3/2 + … + (-81/16)
S = 231 – (-3/2)6
1 – (-3/2)
= 23
1 – (729/64)1 + (3/2)
= 23
-665/645/2
-13348
Geometric Sequences
Set a1 = 2/3, r = -3/2 and n = 6 in the formula 1 – rn
1 – rS = a1
we get the sum S
=
Therefore there are 6 terms in the sum, 2/3 + (-1) + 3/2 + … + (-81/16)
S = 231 – (-3/2)6
1 – (-3/2)
= 23
1 – (729/64)1 + (3/2)
= 23
-665/645/2
-13348
Geometric Sequences
Set a1 = 2/3, r = -3/2 and n = 6 in the formula 1 – rn
1 – rS = a1
we get the sum S
=
Therefore there are 6 terms in the sum, 2/3 + (-1) + 3/2 + … + (-81/16)
S = 231 – (-3/2)6
1 – (-3/2)
= 23
1 – (729/64)1 + (3/2)
= 23
-665/645/2
-13348
Geometric Sequences
Set a1 = 2/3, r = -3/2 and n = 6 in the formula 1 – rn
1 – rS = a1
we get the sum S
=
Geometric SequencesHW. Given that a1, a2 , a3 , …is a geometric sequence find a1, r, and the specific formula for the an.1. a2 = 15, a5 = 405
2. a3 = 3/4, a6 = –2/9
2. a4 = –5/2, a8 = –40
Sum the following geometric sequences.1. 3 + 6 + 12 + .. + 3072
1. –2 + 6 –18 + .. + 486
1. 6 – 3 + 3/2 – .. + 3/512
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