Question

What graph look like if 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 graph the given geometric sequence, we first need to understand how the values are changing. The provided table has two columns: \( n \) (the index) and \( f(n) \) (the value of the sequence at that index).

Here's the data from the table:

\[
\begin{align*}
n & \quad f(n) \\
0 & \quad 11 \\
1 & \quad 16.5 \\
2 & \quad 24.75 \\
3 & \quad 37.125 \\
4 & \quad 55.6875 \\
\end{align*}
\]

### Step 1: Understanding the Sequence
To understand if it's a geometric sequence, we can check if there is a constant ratio between consecutive terms.

Calculating the ratio between consecutive terms:

- For \( n=0 \) and \( n=1 \):
\[
r_1 = \frac{f(1)}{f(0)} = \frac{16.5}{11} = 1.5
\]

- For \( n=1 \) and \( n=2 \):
\[
r_2 = \frac{f(2)}{f(1)} = \frac{24.75}{16.5} = 1.5
\]

- For \( n=2 \) and \( n=3 \):
\[
r_3 = \frac{f(3)}{f(2)} = \frac{37.125}{24.75} = 1.5
\]

- For \( n=3 \) and \( n=4 \):
\[
r_4 = \frac{f(4)}{f(3)} = \frac{55.6875}{37.125} = 1.5
\]

Since the ratio is constant \( r = 1.5 \), this confirms it's a geometric sequence.

### Step 2: Graphing the Sequence
In order to graph \( f(n) \):

1. **Plotting Points:**
You would plot the points corresponding to the values in the table:
- (0, 11)
- (1, 16.5)
- (2, 24.75)
- (3, 37.125)
- (4, 55.6875)

2. **Drawing the Graph:**
- The x-axis represents \( n \) and the y-axis represents \( f(n) \).
- As \( n \) increases, \( f(n) \) increases following the ratio of 1.5, which indicates an exponential growth pattern characteristic of geometric sequences.

### Visual Characteristics:
- The graph will start at (0, 11) and will rise steeply as \( n \) increases, reflecting the multiplicative growth (as \( n \) increases, the values of \( f(n) \) will grow exponentially).

### Summary
When you plot the above points, you will see a curve that rises steeply, demonstrating the exponential growth of the geometric sequence based on the constant ratio of 1.5 between successive values.