Let's evaluate the sums of the given sequences step-by-step:
### First Sequence: [tex]\(1, \frac{1}{2}, \frac{1}{4}, \frac{1}{8}\)[/tex]
This is a geometric sequence where:
- The first term [tex]\(a_1\)[/tex] is [tex]\(1\)[/tex].
- The common ratio [tex]\(r_1\)[/tex] is [tex]\(\frac{1}{2}\)[/tex].
To find the sum of an infinite geometric series, we use the formula:
[tex]\[ S = \frac{a}{1 - r} \][/tex]
For our sequence:
- [tex]\(a_1 = 1\)[/tex]
- [tex]\(r_1 = \frac{1}{2}\)[/tex]
Plugging these values into the formula, we get:
[tex]\[ S_1 = \frac{1}{1 - \frac{1}{2}} = \frac{1}{\frac{1}{2}} = 2 \][/tex]
Thus, the sum of the first sequence is:
[tex]\[ S_1 = 2.0 \][/tex]
### Second Sequence: [tex]\(8, 4, 2, 1, \frac{1}{2}, \frac{1}{4}\)[/tex]
This is another geometric sequence where:
- The first term [tex]\(a_2\)[/tex] is [tex]\(8\)[/tex].
- The common ratio [tex]\(r_2\)[/tex] is also [tex]\(\frac{1}{2}\)[/tex].
Using the same formula for the sum of an infinite geometric series:
[tex]\[ S = \frac{a}{1 - r} \][/tex]
For this sequence:
- [tex]\(a_2 = 8\)[/tex]
- [tex]\(r_2 = \frac{1}{2}\)[/tex]
Plugging these values in, we get:
[tex]\[ S_2 = \frac{8}{1 - \frac{1}{2}} = \frac{8}{\frac{1}{2}} = 16 \][/tex]
Thus, the sum of the second sequence is:
[tex]\[ S_2 = 16.0 \][/tex]
### Summary
- The sum of the sequence [tex]\(1, \frac{1}{2}, \frac{1}{4}, \frac{1}{8}\)[/tex] is [tex]\(2.0\)[/tex].
- The sum of the sequence [tex]\(8, 4, 2, 1, \frac{1}{2}, \frac{1}{4}\)[/tex] is [tex]\(16.0\)[/tex].
