Certainly! Let's analyze the sequence step-by-step to find the appropriate recursive formula.
Given the sequence:
[tex]\[ 9.6, -4.8, 2.4, -1.2, 0.6, \ldots \][/tex]
1. Identify the First Term:
The first term, \( f(1) \), is given as \( 9.6 \).
2. Determine the Ratio Between Successive Terms:
We can check the ratio between each successive term to see if there is a common ratio.
[tex]\[
\frac{-4.8}{9.6} = -0.5
\][/tex]
[tex]\[
\frac{2.4}{-4.8} = -0.5
\][/tex]
[tex]\[
\frac{-1.2}{2.4} = -0.5
\][/tex]
[tex]\[
\frac{0.6}{-1.2} = -0.5
\][/tex]
Each term is obtained by multiplying the previous term by \(-0.5\). Thus, the ratio between successive terms is consistently \(-0.5\).
3. Formulate the Recursive Relation:
Given the common ratio of \(-0.5\), the next term in the sequence can be written in terms of the previous term as:
[tex]\[
f(n+1) = (-0.5) \cdot f(n)
\][/tex]
4. Align with Given Options:
Now, let's match this with the provided options:
- \(\mu(n+1) = (-0.5)(n)\): This option does not use the previous term to calculate the next term.
- \(f(n+1) = (0.5)f(n)\): This suggests a positive ratio, which does not match our ratio of \(-0.5\).
- \(f(n+1) = f(0.5n)\): This option incorrectly uses \(0.5n\) instead of \(-0.5 \cdot f(n)\).
- \(f(n+1) = f(-0.5n)\): This option incorrectly positions the ratio inside the function argument.
The correct recursive relation based on our calculation is:
[tex]\[
f(n+1) = (-0.5) \cdot f(n)
\][/tex]
So, the appropriate recursive formula to generate this sequence is:
[tex]\[ f(n+1) = (-0.5) \cdot f(n) \][/tex]
The answer is clearly:
[tex]\[ 2 \][/tex]
