Select the correct answer.

Which expression represents the series [tex]\(1+5+25+125+625\)[/tex]?

A. [tex]\(\sum_{i=0}^4 5^i\)[/tex]

B. [tex]\(\sum_{i=1}^4 5^i\)[/tex]

C. [tex]\(\sum_{i=0}^5 5^i\)[/tex]

D. [tex]\(\sum_{i=1}^5 5^i\)[/tex]



Answer :

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]