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Section exercises

Verbal

What is a geometric sequence?

A sequence in which the ratio between any two consecutive terms is constant.

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How is the common ratio of a geometric sequence found?

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What is the procedure for determining whether a sequence is geometric?

Divide each term in a sequence by the preceding term. If the resulting quotients are equal, then the sequence is geometric.

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What is the difference between an arithmetic sequence and a geometric sequence?

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Describe how exponential functions and geometric sequences are similar. How are they different?

Both geometric sequences and exponential functions have a constant ratio. However, their domains are not the same. Exponential functions are defined for all real numbers, and geometric sequences are defined only for positive integers. Another difference is that the base of a geometric sequence (the common ratio) can be negative, but the base of an exponential function must be positive.

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Algebraic

For the following exercises, find the common ratio for the geometric sequence.

1 , 3 , 9 , 27 , 81 , ...

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0.125 , 0.25 , 0.5 , 1 , 2 , ...

The common ratio is 2

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2 , 1 2 , 1 8 , 1 32 , 1 128 , ...

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For the following exercises, determine whether the sequence is geometric. If so, find the common ratio.

6 , 12 , 24 , 48 , 96 , ...

The sequence is geometric. The common ratio is 2.

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5 , 5.2 , 5.4 , 5.6 , 5.8 , ...

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1 , 1 2 , 1 4 , 1 8 , 1 16 , ...

The sequence is geometric. The common ratio is 1 2 .

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6 , 8 , 11 , 15 , 20 , ...

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0.8 , 4 , 20 , 100 , 500 , ...

The sequence is geometric. The common ratio is 5.

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For the following exercises, write the first five terms of the geometric sequence, given the first term and common ratio.

a 1 = 5 , r = 1 5

5 , 1 , 1 5 , 1 25 , 1 125

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For the following exercises, write the first five terms of the geometric sequence, given any two terms.

a 6 = 25 , a 8 = 6.25

800 , 400 , 200 , 100 , 50

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For the following exercises, find the specified term for the geometric sequence, given the first term and common ratio.

The first term is 2, and the common ratio is 3. Find the 5 th term.

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The first term is 16 and the common ratio is 1 3 . Find the 4 th term.

a 4 = 16 27

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For the following exercises, find the specified term for the geometric sequence, given the first four terms.

a n = { 1 , 2 , 4 , 8 , ... } . Find a 12 .

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a n = { 2 , 2 3 , 2 9 , 2 27 , ... } . Find a 7 .

a 7 = 2 729

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For the following exercises, write the first five terms of the geometric sequence.

a 1 = 486 , a n = 1 3 a n 1

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a 1 = 7 , a n = 0.2 a n 1

7 , 1.4 , 0.28 , 0.056 , 0.0112

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For the following exercises, write a recursive formula for each geometric sequence.

a n = { 1 , 5 , 25 , 125 , ... }

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a n = { 32 , 16 , 8 , 4 , ... }

a = 1 32 , a n = 1 2 a n 1

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a n = { 14 , 56 , 224 , 896 , ... }

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a n = { 10 , 3 , 0.9 , 0.27 , ... }

a 1 = 10 , a n = 0.3 a n 1

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a n = { 0.61 , 1.83 , 5.49 , 16.47 , ... }

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a n = { 3 5 , 1 10 , 1 60 , 1 360 , ... }

a 1 = 3 5 , a n = 1 6 a n 1

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a n = { 2 , 4 3 , 8 9 , 16 27 , ... }

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a n = { 1 512 , 1 128 , 1 32 , 1 8 , ... }

a 1 = 1 512 , a n = 4 a n 1

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For the following exercises, write the first five terms of the geometric sequence.

a n = 4 5 n 1

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a n = 12 ( 1 2 ) n 1

12 , 6 , 3 , 3 2 , 3 4

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For the following exercises, write an explicit formula for each geometric sequence.

a n = { 2 , 4 , 8 , 16 , ... }

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a n = { 1 , 3 , 9 , 27 , ... }

a n = 3 n 1

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a n = { 4 , 12 , 36 , 108 , ... }

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a n = { 0.8 , 4 , 20 , 100 , ... }

a n = 0.8 ( 5 ) n 1

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a n = { 1.25 , 5 , 20 , 80 , ... }

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a n = { 1 , 4 5 , 16 25 , 64 125 , ... }

a n = ( 4 5 ) n 1

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a n = { 2 , 1 3 , 1 18 , 1 108 , ... }

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a n = { 3 , 1 , 1 3 , 1 9 , ... }

a n = 3 ( 1 3 ) n 1

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For the following exercises, find the specified term for the geometric sequence given.

Let a 1 = 4 , a n = 3 a n 1 . Find a 8 .

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Let a n = ( 1 3 ) n 1 . Find a 12 .

a 12 = 1 177 , 147

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For the following exercises, find the number of terms in the given finite geometric sequence.

a n = { 1 , 3 , 9 , ... , 2187 }

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a n = { 2 , 1 , 1 2 , ... , 1 1024 }

There are 12 terms in the sequence.

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Graphical

For the following exercises, determine whether the graph shown represents a geometric sequence.

Graph of a scattered plot with labeled points: (1, -0.5), (2, 0.25), (3, 1.375), (4, 3.0625), and (5, 5.5938). The x-axis is labeled n and the y-axis is labeled a_n.

The graph does not represent a geometric sequence.

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For the following exercises, use the information provided to graph the first five terms of the geometric sequence.

a 1 = 3 , a n = 2 a n 1

Graph of a scattered plot with labeled points: (1, 3), (2, 6), (3, 12), (4, 24), and (5, 48). The x-axis is labeled n and the y-axis is labeled a_n.
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a n = 27 0.3 n 1

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Extensions

Use recursive formulas to give two examples of geometric sequences whose 3 rd terms are 200.

Answers will vary. Examples: a 1 = 800 , a n = 0.5 a n 1 and a 1 = 12.5 , a n = 4 a n 1

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Use explicit formulas to give two examples of geometric sequences whose 7 th terms are 1024.

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Find the 5 th term of the geometric sequence { b , 4 b , 16 b , ... } .

a 5 = 256 b

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Find the 7 th term of the geometric sequence { 64 a ( b ) , 32 a ( 3 b ) , 16 a ( 9 b ) , ... } .

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At which term does the sequence { 10 , 12 , 14.4 , 17.28 ,   ... } exceed 100 ?

The sequence exceeds 100 at the 14 th term, a 14 107.

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At which term does the sequence { 1 2187 , 1 729 , 1 243 , 1 81   ... } begin to have integer values?

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For which term does the geometric sequence a n = 36 ( 2 3 ) n 1 first have a non-integer value?

a 4 = 32 3 is the first non-integer value

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Use the recursive formula to write a geometric sequence whose common ratio is an integer. Show the first four terms, and then find the 10 th term.

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Use the explicit formula to write a geometric sequence whose common ratio is a decimal number between 0 and 1. Show the first 4 terms, and then find the 8 th term.

Answers will vary. Example: Explicit formula with a decimal common ratio: a n = 400 0.5 n 1 ; First 4 terms: 400 , 200 , 100 , 50 ; a 8 = 3.125

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Is it possible for a sequence to be both arithmetic and geometric? If so, give an example.

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Questions & Answers

A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?
Aislinn Reply
cm
tijani
what is titration
John Reply
what is physics
Siyaka Reply
A mouse of mass 200 g falls 100 m down a vertical mine shaft and lands at the bottom with a speed of 8.0 m/s. During its fall, how much work is done on the mouse by air resistance
Jude Reply
Can you compute that for me. Ty
Jude
what is the dimension formula of energy?
David Reply
what is viscosity?
David
what is inorganic
emma Reply
what is chemistry
Youesf Reply
what is inorganic
emma
Chemistry is a branch of science that deals with the study of matter,it composition,it structure and the changes it undergoes
Adjei
please, I'm a physics student and I need help in physics
Adjanou
chemistry could also be understood like the sexual attraction/repulsion of the male and female elements. the reaction varies depending on the energy differences of each given gender. + masculine -female.
Pedro
A ball is thrown straight up.it passes a 2.0m high window 7.50 m off the ground on it path up and takes 1.30 s to go past the window.what was the ball initial velocity
Krampah Reply
2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.
Sahid Reply
you have been hired as an espert witness in a court case involving an automobile accident. the accident involved car A of mass 1500kg which crashed into stationary car B of mass 1100kg. the driver of car A applied his brakes 15 m before he skidded and crashed into car B. after the collision, car A s
Samuel Reply
can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?
Joseph Reply
Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time
Joseph
Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing
Joseph
"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true
Ryan
what's motion
Maurice Reply
what are the types of wave
Maurice
answer
Magreth
progressive wave
Magreth
hello friend how are you
Muhammad Reply
fine, how about you?
Mohammed
hi
Mujahid
A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?
yasuo Reply
Who can show me the full solution in this problem?
Reofrir Reply
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Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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