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Answer :
We start with the series:
[tex]$$-5 + 25 - 125 + 625 - 3125.$$[/tex]
### Step 1. Identify the Series Type
This series is geometric because each term is obtained by multiplying the previous term by the same number.
### Step 2. Determine the First Term and Common Ratio
- The first term is
[tex]$$a = -5.$$[/tex]
- To find the common ratio, divide the second term by the first term:
[tex]$$r = \frac{25}{-5} = -5.$$[/tex]
You can verify that the ratio is consistent:
[tex]$$\frac{-125}{25} = -5, \quad \frac{625}{-125} = -5, \quad \frac{-3125}{625} = -5.$$[/tex]
### Step 3. Count the Number of Terms
There are 5 terms in the series.
### Step 4. Write in Summation Notation
For a geometric series, the [tex]$i$[/tex]-th term is given by:
[tex]$$a \cdot r^{\,i-1}.$$[/tex]
Since we have 5 terms, the series can be written as:
[tex]$$\sum_{i=1}^{5} -5(-5)^{\,i-1}.$$[/tex]
### Final Answer
The summation notation for the series is:
[tex]$$\boxed{\sum_{i=1}^{5} -5(-5)^{\,i-1}}.$$[/tex]
[tex]$$-5 + 25 - 125 + 625 - 3125.$$[/tex]
### Step 1. Identify the Series Type
This series is geometric because each term is obtained by multiplying the previous term by the same number.
### Step 2. Determine the First Term and Common Ratio
- The first term is
[tex]$$a = -5.$$[/tex]
- To find the common ratio, divide the second term by the first term:
[tex]$$r = \frac{25}{-5} = -5.$$[/tex]
You can verify that the ratio is consistent:
[tex]$$\frac{-125}{25} = -5, \quad \frac{625}{-125} = -5, \quad \frac{-3125}{625} = -5.$$[/tex]
### Step 3. Count the Number of Terms
There are 5 terms in the series.
### Step 4. Write in Summation Notation
For a geometric series, the [tex]$i$[/tex]-th term is given by:
[tex]$$a \cdot r^{\,i-1}.$$[/tex]
Since we have 5 terms, the series can be written as:
[tex]$$\sum_{i=1}^{5} -5(-5)^{\,i-1}.$$[/tex]
### Final Answer
The summation notation for the series is:
[tex]$$\boxed{\sum_{i=1}^{5} -5(-5)^{\,i-1}}.$$[/tex]
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