Which of the following correctly graphs the geometric sequence?

Question

n	f(n)
0	11
1	16.5
2	24.75
3	37.125
4	55.6875
(1 point)
Responses

[Five points are plotted on the first quadrant of a coordinate plane. The x-axis ranges from 0 to 6 in increments of 1 and the y-axis ranges from 0 to 6000 in increments of 1000. 
[ Image with alt text: Five points are plotted on the first quadrant of a coordinate plane. The x-axis ranges from 0 to 6 in increments of 1 and the y-axis ranges from 0 to 6000 in increments of 1000.

Five points are plotted on the first quadrant of a coordinate plane. The x-axis ranges from 0 to 6 in unit increments and the y-axis ranges from 0 to 60 in increments of 10. 
Image with alt text: Five points are plotted on the first quadrant of a coordinate plane. The x-axis ranges from 0 to 6 in unit increments and the y-axis ranges from 0 to 60 in increments of 10.

Five points are plotted on the first quadrant of a coordinate plane. The x-axis ranges from 0 to 10 in increments of 1 and the y-axis ranges from 0 to 60 in increments of 10. 
Image with alt text: Five points are plotted on the first quadrant of a coordinate plane. The x-axis ranges from 0 to 10 in increments of 1 and the y-axis ranges from 0 to 60 in increments of 10.

Five points are plotted on the first quadrant of a coordinate plane. The x-axis ranges from 0 to 60 in increments of 10 and the y-axis ranges from 0 to 6 in increments of 1. 
Image with alt text: Five points are plotted on the first quadrant of a coordinate plane. The x-axis ranges from 0 to 60 in increments of 10 and the y-axis ranges from 0 to 6 in increments of 1.
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GPT-4o mini · Answer

To determine which graph correctly represents the given geometric sequence, we first need to analyze the values provided in the table.

The values are:

$ f(0) = 11 $
$ f(1) = 16.5 $
$ f(2) = 24.75 $
$ f(3) = 37.125 $
$ f(4) = 55.6875 $

To identify whether this is indeed a geometric sequence, we can check the ratio of successive terms:

$ \frac{f(1)}{f(0)} = \frac{16.5}{11} = 1.5 $
$ \frac{f(2)}{f(1)} = \frac{24.75}{16.5} = 1.5 $
$ \frac{f(3)}{f(2)} = \frac{37.125}{24.75} = 1.5 $
$ \frac{f(4)}{f(3)} = \frac{55.6875}{37.125} = 1.5 $

Since the ratio is consistently $ 1.5 $, we confirm that this is a geometric sequence where the first term $ a = 11 $ and the common ratio $ r = 1.5 $.

Next, we will calculate the values for $ f(n) $ for $ n = 0, 1, 2, 3, 4 $:

$ f(0) = 11 $
$ f(1) = 11 \times 1.5 = 16.5 $
$ f(2) = 16.5 \times 1.5 = 24.75 $
$ f(3) = 24.75 \times 1.5 = 37.125 $
$ f(4) = 37.125 \times 1.5 = 55.6875 $

Now we need to examine the potential graphs provided in your options. Key characteristics of the graphs we are looking for:

The points plotted should correspond to $ (0, 11), (1, 16.5), (2, 24.75), (3, 37.125), (4, 55.6875) $.
The y-values should range appropriately to include the values up to approximately 55.6875.

Let's summarize the graphs based on the ranges:

Graph A: X-axis (0 to 6), Y-axis (0 to 6000) – Too high for y-values.
Graph B: X-axis (0 to 6), Y-axis (0 to 60) – Possible, y-values fit.
Graph C: X-axis (0 to 10), Y-axis (0 to 60) – Possible, y-values fit.
Graph D: X-axis (0 to 60), Y-axis (0 to 6) – Incorrect range for x and y.

From the analysis above, Graph B and Graph C are the most suitable options since they remain within the necessary scale for both x and y values. However, if we have to choose the most appropriate one, Graph B with y ranging from 0 to 60 would likely show all the points more clearly, as the range would provide a better visualization. Therefore, the best choice is:

Graph B.

adaleta · Answer

Use the table to answer the question.

Minutes Messages Sent
0 0
1 3
2 9
3 27
4 81
The table shows how a text message went viral. What point should be graphed to represent the third term in the geometric sequence?

(1 point)
Responses

(3,27)
left parenthesis 3 comma 27 right parenthesis

(4,81)
left parenthesis 4 comma 81 right parenthesis

(3,1)
left parenthesis 3 comma 1 right parenthesis

(2,9)
left parenthesis 2 comma 9 right parenthesis
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GPT-4o mini · Answer

To determine the correct point representing the third term in the geometric sequence from the table you provided, we first need to identify what the "third term" means in this context.

In the table:

At $0$ minutes, $0$ messages were sent.
At $1$ minute, $3$ messages were sent.
At $2$ minutes, $9$ messages were sent.
At $3$ minutes, $27$ messages were sent.
At $4$ minutes, $81$ messages were sent.

The third term refers to the value when $n = 2$ (using zero-based indexing).

Thus, the third term in the sequence corresponds to:

$ n = 2 $
Messages Sent = $ 9 $

Hence, the point representing the third term would be $(2, 9)$.

The correct response is:

(2, 9).

adaleta · Answer

Which exponential function can be used to show the geometric sequence 40, 160, 640, . . . ?(1 point)
Responses

an=40(120)n−1
a subscript n baseline equals 40 left parenthesis 120 right parenthesis superscript n minus 1 baseline

an=40+4(n−1)
a subscript n baseline equals 40 plus 4 left parenthesis n minus 1 right parenthesis

an=40(4)n−1
a subscript n baseline equals 40 left parenthesis 4 right parenthesis superscript n minus 1 baseline

an=4(40)n−1

GPT-4o mini · Answer

To find the exponential function that represents the geometric sequence $40, 160, 640, \ldots$, we first identify the first term and the common ratio.

First Term $a$: The first term of the sequence, when $n=1$, is $40$.
Common Ratio $r$: To find the common ratio, divide the second term by the first term:

\[ r = \frac{160}{40} = 4 \]

and to confirm the common ratio, divide the third term by the second term:

\[ r = \frac{640}{160} = 4 \]

Both calculations confirm that the common ratio is $4$.

The formula for a geometric sequence can be expressed as:

\[ a_n = a \cdot r^{n-1} \]

Where:

$a$ is the first term (which is $40$),
$r$ is the common ratio (which is $4$),
$n$ is the term number.

Thus, the function representing the sequence is:

\[ a_n = 40 \cdot (4)^{n-1} \]

Looking at the provided options, the correct response is:

$a_n = 40(4)^{n-1}$.

adaleta · Answer

Every year that a car is on the road it loses value. If a car loses 15 percent of its value each year and its beginning value is $24,500, what will the value of the car be after five years?(1 point)
Responses

$9240.16
$9240.16

$49,278.25
$49,278.25

$10,870.78
$10,870.78

$12,789.15
$12,789.15

GPT-4o mini · Answer

To find the value of the car after five years, we can use the formula for exponential decay, given that the car loses 15% of its value each year. The formula can be expressed as:

\[ V = P(1 - r)^t \]

Where:

$V$ is the future value of the car,
$P$ is the initial value of the car (in this case, $24,500),
$r$ is the rate of depreciation (in this case, 0.15 for 15%),
$t$ is the number of years (in this case, 5).

Now, we can substitute the values into the formula:

\[ V = 24500(1 - 0.15)^5 \] \[ V = 24500(0.85)^5 \]

Next, we calculate $0.85^5$:

\[ 0.85^5 \approx 0.4437 \quad (\text{calculated}) \]

Now we substitute this back into the formula:

\[ V \approx 24500 \times 0.4437 \approx 10870.78 \]

Thus, the value of the car after five years is approximately $10,870.78.

The correct response is:

$10,870.78.

adaleta · Answer

Write a function to represent the geometric sequence 6, 18, 54, . . . .(1 point)
Responses

f(x)=6(3)x−1
f left parenthesis x right parenthesis equals 6 left parenthesis 3 right parenthesis superscript x minus 1 baseline

f(x)=6(3)x
f left parenthesis x right parenthesis equals 6 left parenthesis 3 right parenthesis superscript x baseline

f(x)=3(6)x
f left parenthesis x right parenthesis equals 3 left parenthesis 6 right parenthesis superscript x baseline

f(x)=3(6)x−1

GPT-4o mini · Answer

To write a function that represents the geometric sequence $6, 18, 54, \ldots$, we first identify the first term and the common ratio:

First Term $a$: The first term of the sequence is $6$.
Common Ratio $r$: To find the common ratio, divide the second term by the first term: \[ r = \frac{18}{6} = 3 \]

We can confirm the common ratio by dividing the third term by the second term: \[ r = \frac{54}{18} = 3 \]

Now we know:

The first term $a = 6$
The common ratio $r = 3$

The formula for a geometric sequence is given by: \[ f(x) = a \cdot r^{x-1} \] where $x$ is the term number.

Substituting $a$ and $r$ into the formula: \[ f(x) = 6 \cdot (3)^{x-1} \]

Now, looking at the answer choices provided:

$f(x) = 6(3)^{x-1}$
$f(x) = 6(3)^{x}$
$f(x) = 3(6)^{x}$
$f(x) = 3(6)^{x-1}$

The correct function that represents the geometric sequence is:

$f(x) = 6(3)^{x-1}$.