The explicit formula is [tex]a_{n-1} = 256 * 0.5^{n-1[/tex] and the sum of the first 25 terms is approximately 512
The given parameters are:
[tex]a_0 = 256[/tex]
[tex]a_{n +1} = 0.5a_n[/tex]
Set n = 0
[tex]a_1 = 0.5a_0[/tex]
Substitute 256 for a0
[tex]a_1 = 0.5 * 256[/tex]
[tex]a_1 = 128[/tex]
This means that the common ratio is 0.5.
The explicit formula is then represented as:
[tex]a_{n-1} = a_0 * r^{n-1[/tex]
This gives
[tex]a_{n-1} = 256 * 0.5^{n-1[/tex]
This is calculated using:
[tex]S_n = \frac{a_0(r^{n-1} -1)}{r - 1}[/tex]
This gives
[tex]S_{25} = \frac{256 * (0.5^{25-1} -1)}{0.5 - 1}[/tex]
Evaluate
[tex]S_{25} = 512[/tex]
Hence, the sum of the first 25 terms is approximately 512
Read more about geometric series at:
https://brainly.com/question/24643676
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