Let's go through each question one by one:
Question 1: You need to identify the correct graph for the geometric sequence defined by the points given in the table.
To determine which graph is correct, we need to recognize that each value in the sequence corresponds to both the index \( n \) and the output \( f(n) \):
- \( n = 0 \), \( f(0) = 11 \)
- \( n = 1 \), \( f(1) = 16.5 \)
- \( n = 2 \), \( f(2) = 24.75 \)
- \( n = 3 \), \( f(3) = 37.125 \)
- \( n = 4 \), \( f(4) = 55.6875 \)
Since the values of \( f(n) \) increase as \( n \) increases, the graph should show an upward trend. Based on the axis ranges, the graph that correctly displays the values and trends is the one with x-axis ranging not too broadly and with y-values that can accommodate the outputs provided.
Without visuals to see the graphs, choose the graph that best fits a positive, increasing trend in the first quadrant.
Question 2: You want to identify the correct point that represents the third term in the sequence. The table shows:
- (0, 0)
- (1, 3)
- (2, 9)
- (3, 27)
- (4, 81)
The third term corresponds to \( n = 3 \) and the output \( f(3) = 27 \).
Therefore, the correct response is:
- \( (3, 27) \)
Question 3: The geometric sequence is \( 40, 160, 640, \ldots \). We can see that each term is multiplied by 4:
- \( 40 \times 4^1 = 160 \)
- \( 40 \times 4^2 = 640 \)
Thus, the exponentially represented function for this sequence in terms of \( n \) can be written as:
- \( a_n = 40(4)^{n-1} \)
So the correct choice is:
- \( a_n = 40(4)^{n-1} \)
Question 4: If a car loses 15% of its value each year, its remaining value after each year is 85%. This can be modeled with the formula:
\[ V(t) = V_0 \cdot (1 - 0.15)^t \]
Where \( V_0 = 24500 \) and \( t = 5 \): \[ V(5) = 24500 \cdot 0.85^5 \]
Calculating this gives: \[ V(5) \approx 24500 \cdot 0.4437 \approx 10870.78 \]
Therefore, the correct response is:
- \( $10,870.78 \)
Question 5: The sequence \( 6, 18, 54, \ldots \) is geometric with a common ratio of 3.
The general form for a geometric sequence is:
- \( f(x) = a \cdot r^x \)
For the given sequence:
- \( a = 6 \)
- \( r = 3 \)
The correct function that represents the sequence based on typical form starting at \( n = 0 \) would be:
- \( f(x) = 6(3)^x \)
However, if the sequence is indexed starting at \( n = 1 \), then the form can also be expressed as:
- \( f(x) = 6(3)^{x-1} \)
Given the options, the correct response is:
- \( f(x) = 6(3)^{x-1} \) (to assume the sequence starts from \( n=1 \))
Feel free to ask if you need further explanation or assistance!