The $300$ people who were lined up outside the amusement park had all entered by $8$:$15$ a.m. By $8$:$30$, $550$ more had entered. The number who entered each $15$ minutes increased steadily by $250$ after that. All of the people were planning to stay for the parade at noon.

1. How many people entered between $11$:$45$ and noon, just in time for the parade?

   Answer (a):

   $4050$

2. Represent the problem as a series and write a formula for the $n$^th term.

   Hint (b):

   Write the summation representing the total number of people who entered the amusement park to see the parade between $8$:$15$ and $12$:$00$.
   Write the formula for the sequence representing the number of people who entered the park during the n^th quarter-hour.