Suppose Bob challenges himself to do $50$ push$-$ups a day for $60$ days. If he misses the first day, he adjusts his goal to do $51$ push$-$ups the next day and all subsequent days. If he misses the next day also, he adds another push$-$up so that he must do $52$ push$-$ups a day for the remaining $58$ days. This continues until he finally starts exercising and does the required number of push$-$ups for the day. Suppose that Bob never does a day with only a partial number of push$-$ups, and he never does more than are called for. Also, once he finally starts doing push$-$ups, he continues every day until the end of the $60$ days. Assuming Bob does some pushups, what are the maximum and the minimum number of push$-$ups he might end up doing? Prove that your answer is correct