A local food bank has created a unique \(100\)-day plan for collecting canned food donations.
Day 1 Goal: Collect \(50\) cans of food.
Day 2 Goal: Collect \(3\) more cans of food than the current day number plus the same number of cans collected on day \(1\).
Day 3 Goal: Collect \(3\) more cans of food than the current day number plus the same number of cans collected on day \(2\).
Day 4 Goal: Collect \(3\) more cans of food than the day number plus the same number of cans collected on day \(3\).
\(\vdots\)
Day 100 Goal: Collect \(3\) more cans of food than the day number plus the same number of cans collected on day \(99\).
How many cans of food will the food bank collect on the 100th day?
Note:
In solving this problem, it may be helpful to use the fact that the sum
of the first \(n\) positive integers is
equal to \(\dfrac{n(n+1)}{2}\). That
is, \[1 + 2 + 3 + \cdots + n =
\frac{n(n+1)}{2}\]