If f, of, 1, equals, 6f(1)=6 and f, of, n, equals, 5, f, of, n, minus, 1f(n)=5f(n−1) then find the value of f, of, 6f(6).

Given the problem:

- \( f(1) = 6 \)
- \( f(n) = 5f(n-1) \)

We are required to find \( f(6) \).

### Step-by-Step Solution:

1. **Find \( f(2) \):**
   \[
   f(2) = 5f(1) = 5 \times 6 = 30
   \]

2. **Find \( f(3) \):**
   \[
   f(3) = 5f(2) = 5 \times 30 = 150
   \]

3. **Find \( f(4) \):**
   \[
   f(4) = 5f(3) = 5 \times 150 = 750
   \]

4. **Find \( f(5) \):**
   \[
   f(5) = 5f(4) = 5 \times 750 = 3750
   \]

5. **Find \( f(6) \):**
   \[
   f(6) = 5f(5) = 5 \times 3750 = 18750
   \]

Thus, the value of \( f(6) \) is \( \boxed{18750} \).

Would you like more details or have any questions?

Here are some related questions you might find interesting:

1. How would the value of \( f(7) \) be calculated using the same recursive formula?
2. What is the general formula for \( f(n) \) in this sequence?
3. Can the sequence be expressed as a closed-form expression rather than a recursive one?
4. How does the sequence grow as \( n \) increases? Is it linear, exponential, or something else?
5. How would the sequence change if the initial condition \( f(1) \) were different?

**Tip:** In recursive sequences, understanding the base case (like \( f(1) \) here) is crucial for accurately determining subsequent values.

If f(1) = 6 and f(n) = 5f(n−1), then find the value of f(6).

Learn how to calculate the value of f(6) using the recursive formula f(n) = 5f(n-1) with a base value of f(1) = 6. This problem demonstrates the steps to solve a recursive sequence, suitable for students in Grades 9-10.

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