To evaluate [tex]\( S_5 \)[/tex] for the series [tex]\( 400 + 200 + 100 + \ldots \)[/tex] which is a geometric series, we need to determine the sum of the first five terms of the series.
First, identify the key components of the geometric series:
1. First term [tex]\( a \)[/tex]: The first term [tex]\( a \)[/tex] is 400.
2. Common ratio [tex]\( r \)[/tex]: The common ratio [tex]\( r \)[/tex] is determined by the ratio of consecutive terms. In this series, each term is half of the previous term. Thus, [tex]\( r = \frac{1}{2} \)[/tex].
The formula for the sum of the first [tex]\( n \)[/tex] terms of a geometric series is given by:
[tex]\[
S_n = a \frac{1 - r^n}{1 - r}
\][/tex]
For this specific problem, we are looking for [tex]\( S_5 \)[/tex]:
1. Substitute [tex]\( a = 400 \)[/tex].
2. Substitute [tex]\( r = 0.5 \)[/tex].
3. Substitute [tex]\( n = 5 \)[/tex].
Now, apply the values in the formula:
[tex]\[
S_5 = 400 \frac{1 - (0.5)^5}{1 - 0.5}
\][/tex]
Next, evaluate the expression step-by-step:
1. Calculate [tex]\( r^5 = (0.5)^5 = 0.03125 \)[/tex].
2. Compute [tex]\( 1 - r^5 = 1 - 0.03125 = 0.96875 \)[/tex].
3. Calculate the denominator [tex]\( 1 - r = 1 - 0.5 = 0.5 \)[/tex].
4. Substitute and compute the series sum:
[tex]\[
S_5 = 400 \frac{0.96875}{0.5} = 400 \times 1.9375 = 775
\][/tex]
So, the sum of the first five terms of the geometric series [tex]\( 400 + 200 + 100 + \ldots \)[/tex] is:
[tex]\[
S_5 = 775
\][/tex]
The correct answer is [tex]\( 775 \)[/tex].
