Final answer:
The winner will have collected $3750 after 30 months.
Step-by-step explanation:
The winner receives $50 the first month, $75 the second month, $100 the third month, and so on. This is an arithmetic sequence where the common difference between each term is $25. To find the sum of the first 30 terms, we can use the formula for the sum of an arithmetic sequence:
S = (n/2)(2a + (n-1)d)
where S is the sum of the terms, n is the number of terms, a is the first term, and d is the common difference.
Plugging in the values, we have:
- n = 30 (number of terms)
- a = 50 (first term)
- d = 25 (common difference)
Using the formula, we can find the sum of the first 30 terms:
S = (30/2)(2*50 + (30-1)*25) = $3750
Therefore, the winner will have collected $3750 after 30 months.