To determine which graph correctly represents the given geometric sequence, we first need to analyze the sequence values provided:
- \( f(0) = 11 \)
- \( f(1) = 16.5 \)
- \( f(2) = 24.75 \)
- \( f(3) = 37.125 \)
- \( f(4) = 55.6875 \)
Identifying the Pattern:
This series represents a geometric sequence, where each term is multiplied by a common ratio to obtain the next term. To find the common ratio \( r \):
- \( r = \frac{f(1)}{f(0)} = \frac{16.5}{11} = 1.5 \)
- Check for the next terms: \[ f(2) = f(1) \cdot r = 16.5 \cdot 1.5 = 24.75 \] \[ f(3) = f(2) \cdot r = 24.75 \cdot 1.5 = 37.125 \] \[ f(4) = f(3) \cdot r = 37.125 \cdot 1.5 = 55.6875 \]
The terms confirm that the common ratio is indeed \( 1.5 \).
Points to Plot:
Now, we plot these points on a coordinate plane:
- (0, 11)
- (1, 16.5)
- (2, 24.75)
- (3, 37.125)
- (4, 55.6875)
Range of Values:
- x-values range from 0 to 4 (the inputs n).
- y-values range approximately from 11 to 55.6875.
Analyzing the Options:
We need to find the graph that has:
- x-axis covering at least the range 0 to 6.
- y-axis covering the range of at least 0 to 60, since 55.6875 is close to 60.
Considering the provided descriptions:
- First Option: x from 0 to 60 and y from 0 to 6.
- Second Option: x from 0 to 6 and y from 0 to 6000.
- Third Option: x from 0 to 10 and y from 0 to 60.
- Fourth Option: x from 0 to 6 and y from 0 to 60.
Conclusion:
The fourth option, which plots the x's from 0 to 6 and y's from 0 to 60, will correctly show our series as it accommodates the output values of the sequence well.
Thus, the correct answer is:
Five points are plotted on the first quadrant of a coordinate plane. The x-axis ranges from 0 to 6 in unit increments and the y-axis ranges from 0 to 60 in increments of 10.
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