This problem describes a scenario in which a firm starts with a loss in the first month but gradually increases its profit each month. Let’s break it down step by step:

Given:

• Loss in the first month: $-2500$
• Monthly profit increases by $450 per month for 12 months.

The firm experiences increasing profits over the year, starting from $-2500$ in the first month and increasing by $450 each month. This forms an arithmetic sequence where the first term $a_1 = -2500$ and the common difference $d = 450$.

Arithmetic Sequence Formula:

The profit for the $n$-th month can be found using: $a_n = a_1 + (n-1) \cdot d$

To find the total profit or loss over the entire year, we need to sum the profits from month 1 to month 12. The sum $S_n$ of the first $n$ terms of an arithmetic sequence is given by: $S_n = \frac{n}{2} \cdot (a_1 + a_n)$ Where:

• $n = 12$ (the total number of months),
• $a_1 = -2500$,
• $a_n = a_1 + (n-1) \cdot d$.

Let’s calculate this now.The firm's total loss for the first year is $300.

This means that despite increasing profits each month, the firm still experienced an overall loss by the end of the first year.

Would you like further clarification or additional details on how this was calculated?

Here are 5 related questions you could explore:

1. How would the total profit or loss change if the monthly increase was larger or smaller?
2. What would happen if the firm continued the same trend into the second year?
3. How does the arithmetic sequence formula apply to other business models?
4. Could there be a point in future months where the firm becomes profitable? How would we calculate that?
5. How does compound interest or exponential growth differ from this arithmetic growth?

Tip: Arithmetic sequences grow linearly, making them useful for modeling situations with consistent, fixed changes over time.