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Answer :
To express the series [tex]\(5 + 25 + 125 + 625 + 3125\)[/tex] in sigma notation, let's identify the pattern.
1. Look at each term in the series:
- The first term is [tex]\(5\)[/tex].
- The second term is [tex]\(25\)[/tex].
- The third term is [tex]\(125\)[/tex].
- The fourth term is [tex]\(625\)[/tex].
- The fifth term is [tex]\(3125\)[/tex].
2. Recognize the structure of each term:
- Check if there's a common base and exponent pattern.
- Note that:
- [tex]\(5 = 5^1\)[/tex]
- [tex]\(25 = 5^2\)[/tex]
- [tex]\(125 = 5^3\)[/tex]
- [tex]\(625 = 5^4\)[/tex]
- [tex]\(3125 = 5^5\)[/tex]
3. Identify a pattern:
- Each term can be expressed as [tex]\(5^i\)[/tex], where [tex]\(i\)[/tex] denotes the term number, starting from 1.
4. Write the series in sigma notation:
- We have a series where each term is a power of 5, specifically [tex]\(5^i\)[/tex] from [tex]\(i = 1\)[/tex] to [tex]\(i = 5\)[/tex].
5. Sigma notation expression:
- The entire series can be neatly written using sigma notation as:
[tex]\[
\sum_{i=1}^5 5^i
\][/tex]
Thus, the series [tex]\(5 + 25 + 125 + 625 + 3125\)[/tex] can be expressed as [tex]\(\sum_{i=1}^5 5^i\)[/tex] in sigma notation.
1. Look at each term in the series:
- The first term is [tex]\(5\)[/tex].
- The second term is [tex]\(25\)[/tex].
- The third term is [tex]\(125\)[/tex].
- The fourth term is [tex]\(625\)[/tex].
- The fifth term is [tex]\(3125\)[/tex].
2. Recognize the structure of each term:
- Check if there's a common base and exponent pattern.
- Note that:
- [tex]\(5 = 5^1\)[/tex]
- [tex]\(25 = 5^2\)[/tex]
- [tex]\(125 = 5^3\)[/tex]
- [tex]\(625 = 5^4\)[/tex]
- [tex]\(3125 = 5^5\)[/tex]
3. Identify a pattern:
- Each term can be expressed as [tex]\(5^i\)[/tex], where [tex]\(i\)[/tex] denotes the term number, starting from 1.
4. Write the series in sigma notation:
- We have a series where each term is a power of 5, specifically [tex]\(5^i\)[/tex] from [tex]\(i = 1\)[/tex] to [tex]\(i = 5\)[/tex].
5. Sigma notation expression:
- The entire series can be neatly written using sigma notation as:
[tex]\[
\sum_{i=1}^5 5^i
\][/tex]
Thus, the series [tex]\(5 + 25 + 125 + 625 + 3125\)[/tex] can be expressed as [tex]\(\sum_{i=1}^5 5^i\)[/tex] in sigma notation.
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