To determine the correct expression that represents the series [tex]\(1 + 5 + 25 + 125 + 625\)[/tex], we need to analyze the pattern in the series and how it is generated.
Let's break down each term in the series:
- The first term is [tex]\(1\)[/tex], which can be written as [tex]\(5^0\)[/tex].
- The second term is [tex]\(5\)[/tex], which can be written as [tex]\(5^1\)[/tex].
- The third term is [tex]\(25\)[/tex], which can be written as [tex]\(5^2\)[/tex].
- The fourth term is [tex]\(125\)[/tex], which can be written as [tex]\(5^3\)[/tex].
- The fifth term is [tex]\(625\)[/tex], which can be written as [tex]\(5^4\)[/tex].
We can see that the series consists of the powers of 5 starting from [tex]\(5^0\)[/tex] and going up to [tex]\(5^4\)[/tex].
Now, we need to determine the correct summation expression that represents this series.
Let's write the series with summation notation:
[tex]\[
1 + 5 + 25 + 125 + 625 = 5^0 + 5^1 + 5^2 + 5^3 + 5^4
\][/tex]
The summation notation for this series is:
[tex]\[
\sum_{i=0}^4 5^i
\][/tex]
Hence, the correct answer is:
A. [tex]\(\sum_{i=0}^4 5^i\)[/tex]
