Using Summations math I need to do this exercise In a horse race, the prizes were set so that each jockey received $45 less than the previous one. The one who won the race received $360, and the rest in total received the sum of 1,170. He asks, how many were the riders and how much did the last one receive?

Let's assume that the number of riders in the race is represented by 'n', and the amount received by the last rider is denoted by 'a'. Given that the jockey who won the race received $360, we can write the following equation: $$a = 360$$ We are also told that the rest of the riders received a total sum of $1,170. We can represent the sum of the amounts received by the remaining (n-1) riders using a summation notation: $$\sum_{k=1}^{n-1}(a - 45k)= 1170$$ Expanding the summation notation, we have: $$(a - 45(1)) + (a - 45(2)) + (a - 45(3)) +\ldots + (a - 45(n-1)) = 1170$$ Simplifying the equation, we get: $$na - 45(1 + 2 + 3 + \ldots + (n-1)) = 1170$$ The sum of the first (n-1) natural numbers can be represented as: $$1 + 2 + 3 + ... + (n-1) =\frac{(n-1)(n-1+1)}{2}= \frac{n(n-1)}{2}$$ Substituting this back into the equation, we have: $$na - \frac{45n(n-1)}{2} = 1170$$ Simplifying further, we get: $$2na - 45n(n-1) = 2340$$ Since we know that a = 360, we can substitute that value into the equation: $$2(360)n - 45(n-1)n = 2340$$ Simplifying again, we have: $$720n - 45(n-1)n = 2340$$ Expanding and rearranging, we get: $$720n - 45n^2 + 45n = 2340$$ Combining like terms, we have a quadratic equation: $$-45n^2 + 765n - 2340 = 0$$ Dividing the entire equation by -45 to simplify: $$n^2 - 17n + 52 = 0$$ Now, we can solve this quadratic equation for 'n' using factoring, completing the square, or the quadratic formula. Factoring the quadratic equation, we get: $$(n - 4)(n - 13) = 0$$ Setting each factor equal to zero: n - 4 = 0 or n - 13 = 0 Solving these equations, we find two possible solutions for 'n': $$n = 4\:\:,\:\: n = 13$$ Therefore, there are two possible scenarios: Scenario 1: There were 4 riders in the race, and the last rider received $360. Scenario 2: There were 13 riders in the race, and the last rider received $360.