Solved on Jan 19, 2024

Describe the graph of the geometric sequence $640, 160, 40, 10, \ldots$ . Select two: (1) Graph shows exponential growth, (2) Graph appears linear, (3) Domain is natural numbers, (4) Range is natural numbers, (5) Graph shows exponential decay.

STEP 1

Assumptions
1. The given sequence is $640, 160, 40, 10, \ldots$
2. We are to describe the graph of this sequence.
3. We are considering the properties of the graph such as growth/decay, linearity, and the domain and range.

STEP 2

Identify the type of sequence. Since each term is obtained by multiplying the previous term by a constant, this sequence is a geometric sequence.

STEP 3

Determine the common ratio of the sequence by dividing any term by the previous term.
$r = \frac{160}{640} = \frac{1}{4}$

STEP 4

Since the common ratio $r$ is between $0$ and $1$ , the sequence is a decreasing geometric sequence.

STEP 5

Conclude that a decreasing geometric sequence represents exponential decay, not exponential growth.

STEP 6

Determine the domain of the sequence. In a sequence, the domain is the set of all possible positions of terms, which is typically the set of natural numbers or positive integers.

STEP 7

Determine the range of the sequence. The range is the set of values that the sequence takes on. In this case, since the sequence is decreasing and all terms are positive, the range is a subset of the positive real numbers.

STEP 8

Assess whether the graph will appear linear. A linear graph is a straight line, which is not the case for a geometric sequence with a common ratio other than $1$ .

STEP 9

Summarize the properties of the graph of the given geometric sequence: - The graph will show exponential decay because the common ratio is less than $1$ . - The graph will not appear linear because the rate of decrease is not constant in terms of addition or subtraction, but rather multiplication by a fraction. - The domain will be the set of natural numbers because the sequence is indexed by the natural numbers. - The range will not be the set of natural numbers because the values of the sequence are not all natural numbers (they are real numbers and can include fractions).
The correct statements describing how the graph of the given geometric sequence should appear are: - The graph will show exponential decay. - The domain will be the set of natural numbers.

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Describe the graph of the geometric sequence $640, 160, 40, 10, \ldots$ . Select two: (1) Graph shows exponential growth, (2) Graph appears linear, (3) Domain is natural numbers, (4) Range is natural numbers, (5) Graph shows exponential decay.

STEP 1

Assumptions
1. The given sequence is $640, 160, 40, 10, \ldots$
2. We are to describe the graph of this sequence.
3. We are considering the properties of the graph such as growth/decay, linearity, and the domain and range.

STEP 2

Identify the type of sequence. Since each term is obtained by multiplying the previous term by a constant, this sequence is a geometric sequence.

STEP 3

Determine the common ratio of the sequence by dividing any term by the previous term.
$r = \frac{160}{640} = \frac{1}{4}$

STEP 4

Since the common ratio $r$ is between $0$ and $1$ , the sequence is a decreasing geometric sequence.

STEP 5

Conclude that a decreasing geometric sequence represents exponential decay, not exponential growth.

STEP 6

Determine the domain of the sequence. In a sequence, the domain is the set of all possible positions of terms, which is typically the set of natural numbers or positive integers.

STEP 7

Determine the range of the sequence. The range is the set of values that the sequence takes on. In this case, since the sequence is decreasing and all terms are positive, the range is a subset of the positive real numbers.

STEP 8

Assess whether the graph will appear linear. A linear graph is a straight line, which is not the case for a geometric sequence with a common ratio other than $1$ .

STEP 9

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Describe the graph of the geometric sequence 640,160,40,10,…640, 160, 40, 10, \ldots640,160,40,10,…. Select two: (1) Graph shows exponential growth, (2) Graph appears linear, (3) Domain is natural numbers, (4) Range is natural numbers, (5) Graph shows exponential decay.

STEP 1

Assumptions1. The given sequence is 640,160,40,10,…640, 160, 40, 10, \ldots640,160,40,10,…2. We are to describe the graph of this sequence.3. We are considering the properties of the graph such as growth/decay, linearity, and the domain and range.

STEP 2

Identify the type of sequence. Since each term is obtained by multiplying the previous term by a constant, this sequence is a geometric sequence.

STEP 3

Determine the common ratio of the sequence by dividing any term by the previous term.r=160640=14 r = \frac{160}{640} = \frac{1}{4} r=640160​=41​

STEP 4

Since the common ratio rrr is between 000 and 111, the sequence is a decreasing geometric sequence.

STEP 5

Conclude that a decreasing geometric sequence represents exponential decay, not exponential growth.

STEP 6

Determine the domain of the sequence. In a sequence, the domain is the set of all possible positions of terms, which is typically the set of natural numbers or positive integers.

STEP 7

Determine the range of the sequence. The range is the set of values that the sequence takes on. In this case, since the sequence is decreasing and all terms are positive, the range is a subset of the positive real numbers.

STEP 8

Assess whether the graph will appear linear. A linear graph is a straight line, which is not the case for a geometric sequence with a common ratio other than 111.

STEP 9

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Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Describe the graph of the geometric sequence $640, 160, 40, 10, \ldots$ . Select two: (1) Graph shows exponential growth, (2) Graph appears linear, (3) Domain is natural numbers, (4) Range is natural numbers, (5) Graph shows exponential decay.

Assumptions
1. The given sequence is $640, 160, 40, 10, \ldots$
2. We are to describe the graph of this sequence.
3. We are considering the properties of the graph such as growth/decay, linearity, and the domain and range.

Determine the common ratio of the sequence by dividing any term by the previous term.
$r = \frac{160}{640} = \frac{1}{4}$

Since the common ratio $r$ is between $0$ and $1$ , the sequence is a decreasing geometric sequence.

Assess whether the graph will appear linear. A linear graph is a straight line, which is not the case for a geometric sequence with a common ratio other than $1$ .