Anderson earns \[tex]$20 per month for completing chores around the house. Each month, the amount Anderson earns increases by \$[/tex]0.50. The total amount of money Anderson earns from month 3 to month 18 can be represented by this expression: [tex]\sum_{n=3}^{18}[20+(n-1) 0.5][/tex]. Which is another way of expressing this amount?

A. [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]

B. [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]

C. [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]

D. None of the above



Answer :

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]