To solve the given problem, we need to identify the nature of the series and determine the correct common ratio.
Given series:
[tex]\[
10000 = \frac{P}{(1+0.04)} + \frac{P}{(1+0.04)^2} + \frac{P}{(1+0.04)^3} + \cdots + \frac{P}{(1+0.04)^{10}}
\][/tex]
Let's analyze the structure of this series:
1. Nature of the Series:
- Each term in the series can be described as [tex]\(\frac{P}{(1+0.04)^n}\)[/tex] where [tex]\(n\)[/tex] is the term number (ranging from 1 to 10).
- Notice that each term after the first is obtained by multiplying the previous term by a common ratio [tex]\(\frac{1}{(1+0.04)}\)[/tex].
2. Common Ratio:
- To identify the common ratio, take the ratio of the second term to the first term:
[tex]\[
\frac{\frac{P}{(1+0.04)^2}}{\frac{P}{(1+0.04)}} = \frac{1}{(1+0.04)}
\][/tex]
- Simplifying further:
[tex]\[
\text{Common ratio} = \frac{1}{1.04}
\][/tex]
Given that [tex]\(\frac{1}{1.04}\)[/tex] simplifies numerically to approximately [tex]\(0.9615384615384615\)[/tex], we can conclude that:
- The series is geometric.
- The common ratio of the series is [tex]\(\frac{1}{1.04}\)[/tex].
Hence, among the given choices, the correct statement is:
- The series is geometric with a common ratio of [tex]\(\frac{1}{1.04}\)[/tex].
