2013 Pascal Problem 25

The *GEB sequence* 1, 3, 7, 12, \ldots is defined by the following properties:

(i) the GEB sequence is increasing (that is, each term is larger than the previous term),

(ii) the sequence formed using the differences between each pair of consecutive terms in the GEB sequence (namely, the sequence 2, 4, 5, \ldots) is increasing, and

(iii) each positive integer that does not occur in the GEB sequence occurs exactly once in the sequence of differences in (ii).

What is the 100th term of the GEB sequence?

\textbf{(A)}\ 5751\quad \textbf{(B)}\ 5724\quad \textbf{(C)}\ 5711\quad \textbf{(D)}\ 5777\quad \textbf{(E)}\ 5764

If there are no answer choices shown, enter a numerical answer.

Full credit to this problem is given to the CEMC, you may view all Pascal contests here.