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

Let's analyze the function \( f(n) \) given by the recursive relation:

\[
f(1) = 5
\]

\[
f(n) = 5 \times f(n-1) \quad \text{for } n > 1
\]

We need to find the value of \( f(6) \).

### Step 1: Calculate the first few terms using the recursive formula

- \( f(1) = 5 \)
- \( f(2) = 5 \times f(1) = 5 \times 5 = 25 \)
- \( f(3) = 5 \times f(2) = 5 \times 25 = 125 \)
- \( f(4) = 5 \times f(3) = 5 \times 125 = 625 \)
- \( f(5) = 5 \times f(4) = 5 \times 625 = 3125 \)
- \( f(6) = 5 \times f(5) = 5 \times 3125 = 15625 \)

### Step 2: Conclusion
So, the value of \( f(6) \) is:

\[
\boxed{15625}
\]

Would you like more details on this solution or have any questions?

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Here are 5 related questions to expand your understanding:

1. How would you describe the general formula for \( f(n) \) given this recursive definition?
2. What is \( f(7) \) using the same pattern?
3. If the recursive relation was \( f(n) = 3 \times f(n-1) \), what would \( f(6) \) be?
4. Can you express \( f(n) \) as a function of \( n \) without using recursion?
5. What happens to \( f(n) \) as \( n \) becomes very large?

**Tip:** Recognizing that recursive functions like this can often be expressed as powers of a base can simplify finding large values.

Learn how to calculate f(6) in the recursive function f(n) = 5 * f(n-1) with a step-by-step solution. This problem involves understanding recursive functions and multiplication in mathematical contexts. Suitable for students in Grades 9-12.

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