Question 2 A) 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 (3,1) left parenthesis 3 comma 1 right parenthesis (3,27) left parenthesis 3 comma 27 right parenthesis (2,9) left parenthesis 2 comma 9 right parenthesis Question 3 A)Which exponential function can be used to show the geometric sequence 40, 160, 640, . . . ?(1 point) Responses 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 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=40+4(n−1) a subscript n baseline equals 40 plus 4 left parenthesis n minus 1 right parenthesis Question 4 A)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 $12,789.15 $12,789.15 $49,278.25 $49,278.25 $10,870.78 $10,870.78 $9240.16 $9240.16 Question 5 A)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)=3(6)x−1 f left parenthesis x right parenthesis equals 3 left parenthesis 6 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)=6(3)x−1 f left parenthesis x right parenthesis equals 6 left parenthesis 3 right parenthesis superscript x minus 1 baseline

Question

GPT-4o mini · Answer

Let's answer each question step by step:

Question 2 A)

From the table:

Minutes | Messages Sent
0       | 0
1       | 3
2       | 9
3       | 27
4       | 81

We are looking for the point that represents the third term in the geometric sequence. The third term in the Messages Sent column corresponds to 27, which occurs at 3 minutes.

Therefore, the point to be graphed is (3, 27).

Answer: (3, 27)

Question 3 A)

The geometric sequence given is 40, 160, 640, ...

To find an exponential function, we can look for a common ratio.

$160 / 40 = 4$
$640 / 160 = 4$

Thus, the common ratio is 4.

As a geometric sequence, we can express it in the form:

$ a_n = a \cdot r^{(n-1)} $

Where:

$ a $ = first term (40)
$ r $ = common ratio (4)

Thus the function becomes:

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

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

Question 4 A)

The car's value decreases by 15% each year, meaning it retains 85% of its value. The formula for the value of the car after $ t $ years can be given by:

\[ V(t) = V_0 \cdot (1 - r)^t \] Where:

$ V_0 = 24,500 $
$ r = 0.15 $
$ t = 5 $ years

Substituting these values:

\[ V(5) = 24500 \cdot (0.85)^5 \]

Calculating it:

Calculate $ (0.85)^5 $: \[ (0.85)^5 \approx 0.4437 \]
Multiply $ 24,500 $ by $ 0.4437 $: \[ 24500 \cdot 0.4437 \approx 10870.78 \]

Answer: $10,870.78

Question 5 A)

Given the geometric sequence 6, 18, 54, ..., we first identify the common ratio:

$ 18 / 6 = 3 $
$ 54 / 18 = 3 $

The common ratio is 3, and the first term is 6.

The general form for the nth term (function) is:

$ f(x) = a \cdot r^x $

Where:

$ a = 6 $
$ r = 3 $

Thus:

$ f(x) = 6 \cdot 3^x $

We can express it in terms of $ n - 1 $:

So, the correct expression is:

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

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

Summary of Answers:

(3, 27)
$ a_n = 40(4)^{n-1} $
$10,870.78
$ f(x) = 6(3)^{x-1} $