Let's analyze the sequence given by the formula [tex]\(a_n = \frac{1}{2^{n-1}}\)[/tex] for [tex]\(n \geq 1\)[/tex].
### Step-by-Step Solution:
1. Understanding the Sequence: The sequence [tex]\(a_n\)[/tex] is given by the formula [tex]\(a_n = \frac{1}{2^{n-1}}\)[/tex]. This formula shows that each term is a fraction with 1 in the numerator and [tex]\(2^{n-1}\)[/tex] in the denominator, where [tex]\(n\)[/tex] is the position of the term in the sequence starting from [tex]\(n = 1\)[/tex].
2. Calculating the First Term ([tex]\(a_1\)[/tex]): For [tex]\(n = 1\)[/tex]: [tex]\[
a_1 = \frac{1}{2^{1-1}} = \frac{1}{2^0} = \frac{1}{1} = 1.0
\][/tex]
3. Calculating the Second Term ([tex]\(a_2\)[/tex]): For [tex]\(n = 2\)[/tex]: [tex]\[
a_2 = \frac{1}{2^{2-1}} = \frac{1}{2^1} = \frac{1}{2} = 0.5
\][/tex]
4. Calculating the Third Term ([tex]\(a_3\)[/tex]): For [tex]\(n = 3\)[/tex]: [tex]\[
a_3 = \frac{1}{2^{3-1}} = \frac{1}{2^2} = \frac{1}{4} = 0.25
\][/tex]
### Summary of Results: - The first term [tex]\(a_1 = 1.0\)[/tex] - The second term [tex]\(a_2 = 0.5\)[/tex] - The third term [tex]\(a_3 = 0.25\)[/tex]
Thus, evaluating the sequence for [tex]\(n = 1\)[/tex], [tex]\(n = 2\)[/tex], and [tex]\(n = 3\)[/tex], we get the terms [tex]\(a_1 = 1.0\)[/tex], [tex]\(a_2 = 0.5\)[/tex], and [tex]\(a_3 = 0.25\)[/tex].
