To determine which sequence can be generated from the formula [tex]\( f(x+1) = \frac{1}{2} f(x) \)[/tex], let's analyze the patterns given:
1. [tex]\( x, \frac{x}{2}, \frac{x}{4}, \frac{x}{6}, \ldots \)[/tex]
2. [tex]\( x, 2x, 4x, 8x, \ldots \)[/tex]
3. [tex]\( x, \frac{x}{2}, \frac{x}{4}, \frac{x}{8}, \ldots \)[/tex]
4. [tex]\( x, 2x, 4x, 6x, \ldots \)[/tex]
### Step-by-Step Analysis:
#### Option 1: [tex]\( x, \frac{x}{2}, \frac{x}{4}, \frac{x}{6}, \ldots \)[/tex]
- The terms in the sequence [tex]\( x, \frac{x}{2}, \frac{x}{4}, \frac{x}{6}, \ldots\)[/tex] follow a pattern for the numerator, but the denominators do not follow a consistent multiplicative pattern (i.e., 6 is not 2 times 4).
#### Option 2: [tex]\( x, 2x, 4x, 8x, \ldots \)[/tex]
- In this sequence, each term is multiplied by 2. However, our formula indicates that each term should be divided by 2, not multiplied.
#### Option 3: [tex]\( x, \frac{x}{2}, \frac{x}{4}, \frac{x}{8}, \ldots \)[/tex]
- This sequence shows that each term is indeed a fraction of [tex]\( x \)[/tex], with the denominator doubling each term, which matches the behavior described by [tex]\( f(x+1) = \frac{1}{2} f(x) \)[/tex]. Specifically:
- Starting with [tex]\( f(x) = x \)[/tex]
- [tex]\( f(x+1) = \frac{x}{2} = \frac{1}{2} x \)[/tex]
- [tex]\( f(x+2) = \frac{1}{2} \left(\frac{x}{2}\right) = \frac{x}{4} \)[/tex]
- [tex]\( f(x+3) = \frac{1}{2} \left(\frac{x}{4}\right) = \frac{x}{8} \)[/tex]
- and so forth.
#### Option 4: [tex]\( x, 2x, 4x, 6x, \ldots \)[/tex]
- This sequence increases by a constant multiple of the initial term with an additive pattern (i.e., multiplying by 2, adding 2x, etc.), which does not align with the given formula of halving each term.
Based on the analysis, the correct sequence that matches the formula [tex]\( f(x+1) = \frac{1}{2} f(x) \)[/tex] is:
[tex]\[
x, \frac{x}{2}, \frac{x}{4}, \frac{x}{8}, \ldots
\][/tex]
Thus, the correct answer is:
[tex]\[ x, \frac{x}{2}, \frac{x}{4}, \frac{x}{8}, \ldots \][/tex]
