Question

a sequence is defined by the recursive function \\(f(n + 1) = \frac{1}{3}f(n)\\). if \\(f(3) = 9\\), what is \\(f(1)\\)?

1
3
27
81

Explanation:

Identify the given recursive relation

We are given a sequence defined by the recursive function:

$$f(n + 1) = \frac{1}{3}f(n)$$

We are also given the value of a specific term in the sequence:

$$f(3) = 9$$

Our goal is to find the value of the first term, \(f(1)\).

Relate the terms using the recursive formula

The recursive formula allows us to find a term if we know the preceding term. Let's write down the relationship for \(n = 2\):

$$f(3) = \frac{1}{3}f(2)$$

We can rearrange this equation to solve for \(f(2)\) in terms of \(f(3)\):

$$f(2) = 3 \cdot f(3)$$

Calculate the value of the second term

Substitute the given value \(f(3) = 9\) into our rearranged equation:

$$f(2) = 3 \cdot 9 = 27$$

Relate the second term to the first term

Now, let's write down the recursive relationship for \(n = 1\):

$$f(2) = \frac{1}{3}f(1)$$

Rearranging this equation to solve for \(f(1)\) gives:

$$f(1) = 3 \cdot f(2)$$

Calculate the value of the first term

Substitute the value of \(f(2) = 27\) into the equation:

$$f(1) = 3 \cdot 27 = 81$$

Answer:

• 1
• 3
• 27
• 81 (Correct answer)