To determine the correct graph of the geometric sequence given the values of \( n \) and \( f(n) \), we start by observing the values provided:
- When \( n = 0 \), \( f(0) = 11 \)
- When \( n = 1 \), \( f(1) = 16.5 \)
- When \( n = 2 \), \( f(2) = 24.75 \)
- When \( n = 3 \), \( f(3) = 37.125 \)
- When \( n = 4 \), \( f(4) = 55.6875 \)
Step 1: Calculate the common ratio
To identify if this is a geometric sequence, we can calculate the ratio between consecutive terms:
\[ r = \frac{f(n+1)}{f(n)} \]
Calculating the ratios:
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From \( f(0) \) to \( f(1) \): \[ r = \frac{16.5}{11} \approx 1.5 \]
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From \( f(1) \) to \( f(2) \): \[ r = \frac{24.75}{16.5} \approx 1.5 \]
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From \( f(2) \) to \( f(3) \): \[ r = \frac{37.125}{24.75} \approx 1.5 \]
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From \( f(3) \) to \( f(4) \): \[ r = \frac{55.6875}{37.125} \approx 1.5 \]
Conclusion
Since the ratio \( r \) is consistent at approximately \( 1.5 \), this confirms that the sequence is indeed geometric.
Step 2: Expressing the function
Given that the first term \( f(0) = 11 \) and the common ratio \( r = 1.5 \), we can express the general term of the geometric sequence as:
\[ f(n) = 11 \times (1.5)^n \]
Plotting the values
If you were to graph \( f(n) \) against \( n \) (where \( n \) is the x-axis and \( f(n) \) is the y-axis), you would see points that start at 11 when \( n=0 \) and increase exponentially as \( n \) increases. The points are roughly calculated as follows:
- \( n = 0, f(0) = 11 \)
- \( n = 1, f(1) = 16.5 \)
- \( n = 2, f(2) = 24.75 \)
- \( n = 3, f(3) = 37.125 \)
- \( n = 4, f(4) = 55.6875 \)
The graph would have a positive curvature, indicating exponential growth.
If you have options for the graph, look for a graph that is steeply increasing and curves upwards, reflecting the exponential nature of the geometric sequence. If you're able to share the options available for the graph, I can assist with identifying which one corresponds to this sequence.