To determine the sum [tex]\(S_5\)[/tex] of the first 5 terms of the geometric series given by [tex]\(300, 150, 75, \ldots\)[/tex], we need to follow these steps:
1. Identify the first term ([tex]\(a\)[/tex]) and the common ratio ([tex]\(r\)[/tex]) of the geometric series:
- The first term ([tex]\(a\)[/tex]) is clearly the first number in the series, which is [tex]\(300\)[/tex].
- To find the common ratio ([tex]\(r\)[/tex]), we divide the second term by the first term:
[tex]\[
r = \frac{150}{300} = 0.5.
\][/tex]
2. Write the formula for the sum of the first [tex]\(n\)[/tex] terms of a geometric series:
- The sum [tex]\(S_n\)[/tex] of the first [tex]\(n\)[/tex] terms of a geometric series can be calculated using the formula:
[tex]\[
S_n = a \frac{1 - r^n}{1 - r},
\][/tex]
where [tex]\(a\)[/tex] is the first term, [tex]\(r\)[/tex] is the common ratio, and [tex]\(n\)[/tex] is the number of terms.
3. Substitute the values [tex]\(a = 300\)[/tex], [tex]\(r = 0.5\)[/tex], and [tex]\(n = 5\)[/tex] into the formula:
[tex]\[
S_5 = 300 \cdot \frac{1 - (0.5)^5}{1 - 0.5}.
\][/tex]
4. Simplify the expression step-by-step:
[tex]\[
S_5 = 300 \cdot \frac{1 - (0.5)^5}{0.5}.
\][/tex]
[tex]\[
(0.5)^5 = 0.5 \cdot 0.5 \cdot 0.5 \cdot 0.5 \cdot 0.5 = 0.03125.
\][/tex]
[tex]\[
S_5 = 300 \cdot \frac{1 - 0.03125}{0.5}.
\][/tex]
[tex]\[
1 - 0.03125 = 0.96875.
\][/tex]
[tex]\[
S_5 = 300 \cdot \frac{0.96875}{0.5}.
\][/tex]
[tex]\[
\frac{0.96875}{0.5} = 1.9375.
\][/tex]
[tex]\[
S_5 = 300 \cdot 1.9375 = 581.25.
\][/tex]
5. Conclusion:
The sum [tex]\(S_5\)[/tex] of the first [tex]\(5\)[/tex] terms of the geometric series [tex]\(300, 150, 75, \ldots\)[/tex] is [tex]\(581.25\)[/tex].
The correct answer is:
[tex]\[
\boxed{581.25}
\][/tex]
