To understand which sequence is generated by the formula [tex]\( f(x+1) = \frac{1}{2} f(x) \)[/tex], let's evaluate the first few terms step-by-step, starting with an initial term [tex]\( f(0) = x \)[/tex].
1. First term:
[tex]\[ f(0) = x \][/tex]
2. Second term:
[tex]\[ f(1) = \frac{1}{2} f(0) = \frac{x}{2} \][/tex]
3. Third term:
[tex]\[ f(2) = \frac{1}{2} f(1) = \frac{1}{2} \left( \frac{x}{2} \right) = \frac{x}{4} \][/tex]
4. Fourth term:
[tex]\[ f(3) = \frac{1}{2} f(2) = \frac{1}{2} \left( \frac{x}{4} \right) = \frac{x}{8} \][/tex]
By continuing this pattern, we can see that each term is half of the preceding term. The sequence generated is:
[tex]\[ x, \frac{x}{2}, \frac{x}{4}, \frac{x}{8}, \ldots \][/tex]
Therefore, the sequence corresponding to the given 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]
So, the correct answer is:
[tex]\[ x, \frac{x}{2}, \frac{x}{4}, \frac{x}{8}, \ldots \][/tex]
