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Sagot :
To determine another way of expressing the total amount Anderson earns from month 3 to month 18, we need to analyze and transform the given series expression into different equivalent forms.
The given expression is:
[tex]\[ \sum_{n=3}^{18} [20 + (n-1) \cdot 0.5] \][/tex]
This expression represents the sum of Anderson's earnings from month 3 to month 18 where his monthly earnings increase starting from [tex]$20 and increase by $[/tex]0.50 each month.
Let's examine the other expressions provided to see which one can be equivalent.
### First Expression to Validate
[tex]\[ \sum_{n=1}^{18} 20 + 0.5 \sum_{n=1}^{18} n - 0.5 \sum_{n=1}^{18} 1 - \left( \sum_{n=1}^2 20 + 0.5 \sum_{n=1}^2 n - 0.5 \sum_{n=1}^2 1 \right) \][/tex]
To break it down,
[tex]\[ \sum_{n=1}^{18} 20 \quad + \quad 0.5 \sum_{n=1}^{18} n \quad - \quad 0.5 \sum_{n=1}^{18} 1 \quad - \left( \sum_{n=1}^2 20 \quad + \quad 0.5 \sum_{n=1}^2 n \quad - \quad 0.5 \sum_{n=1}^2 1 \right) \][/tex]
Total earnings from month 1 to 18 minus the total earnings from month 1 to 2. This expression calculates what remains from the earnings of month 3 to 18. It is equivalent to the given series expression.
### Second Expression to Validate
[tex]\[ \sum_{n=1}^{18} 20 + 0.5 \sum_{n=1}^{18} n - 0.5 \sum_{n=1}^{18} 1 - \left( \sum_{n=1}^3 20 + 0.5 \sum_{n=1}^3 n - 0.5 \sum_{n=1}^3 1 \right) \][/tex]
In this case,
[tex]\[ \sum_{n=1}^{18} 20 \quad + \quad 0.5 \sum_{n=1}^{18} n \quad - \quad 0.5 \sum_{n=1}^{18} 1 \quad - \left( \sum_{n=1}^3 20 \quad + \quad 0.5 \sum_{n=1}^3 n \quad - \quad 0.5 \sum_{n=1}^3 1 \right) \][/tex]
Total earnings from month 1 to 18 minus the total earnings from month 1 to 3. This expression calculates what remains from the earnings of month 4 to 18, which is not what we need.
### Third Expression to Validate
[tex]\[ \sum_{n=1}^{18} 20 + 0.5 \sum_{n=1}^{18} n - \left( \sum_{n=1}^2 20 + 0.5 \sum_{n=1}^2 n \right) \][/tex]
Here,
[tex]\[ \sum_{n=1}^{18} 20 \quad + \quad 0.5 \sum_{n=1}^{18} n \quad - \left( \sum_{n=1}^2 20 \quad + \quad 0.5 \sum_{n=1}^2 n \right) \][/tex]
This is again the total earnings from month 1 to 18 minus the total earnings from month 1 to 2. This expression is also equivalent to the given series.
Of these, the correct way of expressing the given amount is:
[tex]\[ \sum_{n=1}^{18} 20 + 0.5 \sum_{n=1}^{18} n - \left( \sum_{n=1}^2 20 + 0.5 \sum_{n=1}^2 n \right) \][/tex]
This third expression matches the solution derived by analyzing the provided series and thus is another way of expressing the total amount Anderson earns from month 3 to month 18:
[tex]\[ \boxed{\sum_{n=1}^{18} 20 + 0.5 \sum_{n=1}^{18} n - \left( \sum_{n=1}^2 20 + 0.5 \sum_{n=1}^2 n \right)} \][/tex]
The given expression is:
[tex]\[ \sum_{n=3}^{18} [20 + (n-1) \cdot 0.5] \][/tex]
This expression represents the sum of Anderson's earnings from month 3 to month 18 where his monthly earnings increase starting from [tex]$20 and increase by $[/tex]0.50 each month.
Let's examine the other expressions provided to see which one can be equivalent.
### First Expression to Validate
[tex]\[ \sum_{n=1}^{18} 20 + 0.5 \sum_{n=1}^{18} n - 0.5 \sum_{n=1}^{18} 1 - \left( \sum_{n=1}^2 20 + 0.5 \sum_{n=1}^2 n - 0.5 \sum_{n=1}^2 1 \right) \][/tex]
To break it down,
[tex]\[ \sum_{n=1}^{18} 20 \quad + \quad 0.5 \sum_{n=1}^{18} n \quad - \quad 0.5 \sum_{n=1}^{18} 1 \quad - \left( \sum_{n=1}^2 20 \quad + \quad 0.5 \sum_{n=1}^2 n \quad - \quad 0.5 \sum_{n=1}^2 1 \right) \][/tex]
Total earnings from month 1 to 18 minus the total earnings from month 1 to 2. This expression calculates what remains from the earnings of month 3 to 18. It is equivalent to the given series expression.
### Second Expression to Validate
[tex]\[ \sum_{n=1}^{18} 20 + 0.5 \sum_{n=1}^{18} n - 0.5 \sum_{n=1}^{18} 1 - \left( \sum_{n=1}^3 20 + 0.5 \sum_{n=1}^3 n - 0.5 \sum_{n=1}^3 1 \right) \][/tex]
In this case,
[tex]\[ \sum_{n=1}^{18} 20 \quad + \quad 0.5 \sum_{n=1}^{18} n \quad - \quad 0.5 \sum_{n=1}^{18} 1 \quad - \left( \sum_{n=1}^3 20 \quad + \quad 0.5 \sum_{n=1}^3 n \quad - \quad 0.5 \sum_{n=1}^3 1 \right) \][/tex]
Total earnings from month 1 to 18 minus the total earnings from month 1 to 3. This expression calculates what remains from the earnings of month 4 to 18, which is not what we need.
### Third Expression to Validate
[tex]\[ \sum_{n=1}^{18} 20 + 0.5 \sum_{n=1}^{18} n - \left( \sum_{n=1}^2 20 + 0.5 \sum_{n=1}^2 n \right) \][/tex]
Here,
[tex]\[ \sum_{n=1}^{18} 20 \quad + \quad 0.5 \sum_{n=1}^{18} n \quad - \left( \sum_{n=1}^2 20 \quad + \quad 0.5 \sum_{n=1}^2 n \right) \][/tex]
This is again the total earnings from month 1 to 18 minus the total earnings from month 1 to 2. This expression is also equivalent to the given series.
Of these, the correct way of expressing the given amount is:
[tex]\[ \sum_{n=1}^{18} 20 + 0.5 \sum_{n=1}^{18} n - \left( \sum_{n=1}^2 20 + 0.5 \sum_{n=1}^2 n \right) \][/tex]
This third expression matches the solution derived by analyzing the provided series and thus is another way of expressing the total amount Anderson earns from month 3 to month 18:
[tex]\[ \boxed{\sum_{n=1}^{18} 20 + 0.5 \sum_{n=1}^{18} n - \left( \sum_{n=1}^2 20 + 0.5 \sum_{n=1}^2 n \right)} \][/tex]
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