https://file.aiquickdraw.com/mathbot/file/1726860125549hq7rqatz.jpg

Let's analyze the problem from the image.

We are given a geometric sequence with the following data points:

- \( f(1) = 2 \)
- \( f(2) \) = ?
- \( f(3) \) = ?
- \( f(4) \) = ?
- \( f(5) = 162 \)

In a geometric sequence, each term is obtained by multiplying the previous term by a constant ratio, \( r \). 

To solve for the missing values and find the explicit and recursive equations:

### Step 1: Use the known terms to find the ratio
We know that:
\[
f(n) = f(1) \cdot r^{n-1}
\]
We have \( f(1) = 2 \) and \( f(5) = 162 \). Substituting these values into the formula for \( f(n) \):
\[
162 = 2 \cdot r^{5-1} = 2 \cdot r^4
\]
Solve for \( r \):
\[
r^4 = \frac{162}{2} = 81
\]
\[
r = \sqrt[4]{81} = 3
\]

### Step 2: Find the missing terms
Now that we know the common ratio \( r = 3 \), we can calculate the missing terms.

- \( f(2) = f(1) \cdot r = 2 \cdot 3 = 6 \)
- \( f(3) = f(2) \cdot r = 6 \cdot 3 = 18 \)
- \( f(4) = f(3) \cdot r = 18 \cdot 3 = 54 \)

So, the missing terms are:
- \( f(2) = 6 \)
- \( f(3) = 18 \)
- \( f(4) = 54 \)

### Step 3: Write the explicit and recursive equations

#### Explicit equation:
The general formula for a geometric sequence is:
\[
f(n) = f(1) \cdot r^{n-1}
\]
Substituting the known values:
\[
f(n) = 2 \cdot 3^{n-1}
\]

#### Recursive equation:
In a recursive form, the formula is:
\[
f(n+1) = f(n) \cdot r
\]
Substitute \( r = 3 \):
\[
f(n+1) = f(n) \cdot 3
\]
with the initial condition \( f(1) = 2 \).

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Let me know if you'd like more details, or if you have any other questions!

Here are five questions you could explore further:
1. What if the common ratio \( r \) was a fraction?
2. How do geometric sequences differ from arithmetic sequences?
3. How can you graph a geometric sequence?
4. What happens when \( r \) is negative in a geometric sequence?
5. How would you solve if \( f(n) \) involved more complex equations?

**Tip**: Always use the known terms of a sequence to solve for the common ratio when working with geometric progressions.

Find the missing terms of the geometric sequence represented in the table, and then write the explicit and recursive equations for the sequence.

Learn how to find the missing terms in a geometric sequence and derive the explicit and recursive equations. This example uses a sequence where f(1) = 2 and f(5) = 162, teaching key concepts such as common ratios and sequence formulas.

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