Use the table to answer the question.

Question

x y
2 a
4 b
Use the arithmetic sequence formula an=−8+(n−1)⋅2
to find the missing values in the table.

(1 point)
Responses

a=−6
; b=−2
a equals negative 6 ; b equals negative 2

a=−15
; b=−13
a equals negative 15 ; b equals negative 13

a=−10
; b=−8
a equals negative 10 ; b equals negative 8

a=−12
; b=−10
a equals negative 12 ; b equals negative 10

Answer 1

To find the values of \( a \) and \( b \) in the table using the arithmetic sequence formula \( a_n = -8 + (n - 1) \cdot 2 \), we can identify the positions \( n \) for \( a \) and \( b \):

1. For \( x = 2 \) (which corresponds to \( a \)):

   • Here, \( n = 2 \).
   • Using the arithmetic sequence formula: \[ a = a_2 = -8 + (2 - 1) \cdot 2 = -8 + 1 \cdot 2 = -8 + 2 = -6 \]

2. For \( x = 4 \) (which corresponds to \( b \)):

   • Here, \( n = 4 \).
   • Using the arithmetic sequence formula: \[ b = a_4 = -8 + (4 - 1) \cdot 2 = -8 + 3 \cdot 2 = -8 + 6 = -2 \]

Thus, the missing values are:

• \( a = -6 \)
• \( b = -2 \)

The correct response is: a = -6; b = -2.

Answer 2

Alex and Jesse are baking holiday muffins. On day one, they baked five muffins, on day two, they baked 11 muffins, and on day three, they baked 17 muffins. What is the slope for this arithmetic sequence?(1 point)
Responses

6
6

1
1

−6
negative 6

−1

Answer 3

To find the slope of the arithmetic sequence formed by the number of muffins baked each day, we can look at the change in the number of muffins baked from one day to the next.

The number of muffins baked over the three days is as follows:

• Day 1: 5 muffins
• Day 2: 11 muffins
• Day 3: 17 muffins

Now, let's calculate the differences between the days:

1. From Day 1 to Day 2: \[ 11 - 5 = 6 \]

2. From Day 2 to Day 3: \[ 17 - 11 = 6 \]

Since the difference is the same (6 muffins) for both intervals, the slope of this arithmetic sequence is 6.

Thus, the correct response is: 6.

Answer 4

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

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

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

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

(2,9)

Answer 5

In the table, the "Minutes" correspond to the input variable (often represented as \( n \)) and the "Messages Sent" corresponds to the output (often represented as \( a_n \)) in the context of a geometric sequence.

The third term occurs when \( n = 3 \):

• From the table, at \( n = 3 \) (which is 3 minutes), the number of messages sent is 27.

Therefore, the point that should be graphed to represent the third term in the geometric sequence is: (3, 27).

Answer 6

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

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

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

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

an=40+4(n−1)

Answer 7

To determine the exponential function that represents the geometric sequence \( 40, 160, 640, \ldots \), we first need to identify the common ratio and express the sequence in terms of the general form for an exponential function.

1. Identify the first term \( a_1 \):

   • \( a_1 = 40 \)

2. Identify the common ratio \( r \):

   • From \( 40 \) to \( 160 \): \[ r = \frac{160}{40} = 4 \]
   • From \( 160 \) to \( 640 \): \[ r = \frac{640}{160} = 4 \]
   • The common ratio is \( r = 4 \).

3. The general form for the nth term of a geometric sequence is given by: \[ a_n = a_1 \cdot r^{n-1} \] where \( a_1 \) is the first term and \( r \) is the common ratio.

4. Plugging in the values:

   • \( a_1 = 40 \)
   • \( r = 4 \)

   So, \[ a_n = 40 \cdot 4^{n-1} \]

Thus, the correct exponential function that can be used to show the geometric sequence \( 40, 160, 640, \ldots \) is:

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

Answer 8

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

$49,278.25
$49,278.25

$12,789.15
$12,789.15

$9240.16
$9240.16

$10,870.78

Answer 9

To determine the value of the car after five years, given that it loses 15% of its value each year, we can use the formula for exponential decay. The formula to calculate the future value \( V \) of an asset after \( t \) years is:

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

Where:

• \( P \) is the initial value (in this case, $24,500),
• \( r \) is the rate of depreciation (15% or 0.15),
• \( t \) is the number of years (in this case, 5 years).

Let's calculate it step-by-step:

1. Calculate \( 1 - r \): \[ 1 - r = 1 - 0.15 = 0.85 \]

2. Calculate \( (1 - r)^t \) for \( t = 5 \): \[ (0.85)^5 \approx 0.4437 \]

3. Calculate the value after 5 years: \[ V = 24500 \times (0.85)^5 = 24500 \times 0.4437 \approx 10870.78 \]

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

The correct response is: $10,870.78.

Answer 10

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

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

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)=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

Answer 11

To find the function that represents the geometric sequence \( 6, 18, 54, \ldots \), we need to determine the first term and the common ratio.

1. Identify the first term \( a_1 \):

   • \( a_1 = 6 \)

2. Identify the common ratio \( r \):

   • From \( 6 \) to \( 18 \): \[ r = \frac{18}{6} = 3 \]
   • From \( 18 \) to \( 54 \): \[ r = \frac{54}{18} = 3 \]
   • The common ratio is \( r = 3 \).

3. The general form for the nth term of a geometric sequence is: \[ f(x) = a_1 \cdot r^x \] or, for it to correspond with typical index notation: \[ f(n) = a_1 \cdot r^{n-1} \]

Because our first term corresponds to \( x = 0 \) in the exponential function setup, if we express the function starting from \( x = 1 \) for the first term, we need to use \( x - 1 \) for the exponent.

So, in our case:

• The function becomes \( f(x) = 6(3)^{x-1} \) if we consider the first term at \( x = 1 \).

Thus, the correct response to represent the geometric sequence \( 6, 18, 54, \ldots \) is: \( f(x) = 6(3)^{x-1} \).