If you have a rewards card at Biggby, you earn a free drink after every 12 drinks you buy. Can the total amount of money spent at Biggby after ordering \( n \) drinks be considered arithmetic? Why or why not?

To determine whether the total amount of money spent at Biggby after ordering \( n \) drinks can be considered arithmetic, we need to analyze how the cost accumulates based on the rewards system. 1. **Cost of Drinks**: Let's assume each drink costs a fixed amount, say \( p \) dollars. 2. **Rewards System**: For every 12 drinks purchased, you earn a free drink. This means that for every 12 drinks, you effectively pay for 12 drinks but receive 13 drinks in total. 3. **Total Cost Calculation**: - If \( n \) is the total number of drinks ordered, we can break it down into how many complete sets of 12 drinks you have: - Let \( k = \left\lfloor \frac{n}{12} \right\rfloor \) (the number of complete sets of 12 drinks). - The number of free drinks you earn is \( k \) (one free drink for each complete set of 12). - The total number of drinks you actually pay for is \( n - k \) (since you get \( k \) drinks for free). 4. **Total Amount Spent**: - The total amount spent can be calculated as: \[ \text{Total Cost} = p \times (n - k) \] - Since \( k = \left\lfloor \frac{n}{12} \right\rfloor \), the expression for total cost becomes: \[ \text{Total Cost} = p \times \left(n - \left\lfloor \frac{n}{12} \right\rfloor\right) \] 5. **Is it Arithmetic?**: - An arithmetic sequence is defined as a sequence of numbers in which the difference between consecutive terms is constant. - The term \( \left\lfloor \frac{n}{12} \right\rfloor \) changes in a stepwise manner. Specifically, it increases by 1 every time \( n \) reaches a multiple of 12. This means that the difference \( n - \left\lfloor \frac{n}{12} \right\rfloor \) does not increase linearly; instead, it has a piecewise linear behavior. - For example: - For \( n = 1 \) to \( n = 11 \), \( k = 0 \) and the total cost is \( pn \). - For \( n = 12 \), \( k = 1 \) and the total cost is \( p(12 - 1) = 11p \). - For \( n = 13 \) to \( n = 23 \), \( k = 1 \) and the total cost is \( p(n - 1) \). - For \( n = 24 \), \( k = 2 \) and the total cost is \( p(24 - 2) = 22p \). - The differences between consecutive total costs are not constant, as they change at multiples of 12. **Conclusion**: The total amount of money spent at Biggby after ordering \( n \) drinks cannot be considered arithmetic because the cost does not increase by a constant amount for each additional drink due to the rewards system. The presence of the floor function and the way free drinks are awarded creates a non-linear accumulation of costs.