Question

Which of the following correctly graphs the geometric sequence?

n f(n)

0 11
1 16.5
2 24.75
3 37.125
4 55.6875

Answer 1

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:

- From \( f(0) \) to \( f(1) \):
\[
r = \frac{16.5}{11} \approx 1.5
\]

- From \( f(1) \) to \( f(2) \):
\[
r = \frac{24.75}{16.5} \approx 1.5
\]

- From \( f(2) \) to \( f(3) \):
\[
r = \frac{37.125}{24.75} \approx 1.5
\]

- 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.