The $G E B$ 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?
(A) 5751
(B) 5724
(C) 5711
(D) 5777
(E) 5764
We refer to the two sequences as the GEB sequence and the difference sequence.

Since the GEB sequence is increasing and since each positive integer that does not occur in the GEB sequence must occur in the difference sequence, then each positive integer less than 12 except 1, 3, 7 (a total of 8 positive integers) must occur in the difference sequence.

Since the difference sequence is increasing, then these 8 positive integers occur in increasing order.

Therefore, the difference sequence begins $2,4,5,6,8,9,10,11, \ldots$

This allows us to continue the GEB sequence using the integers in the difference sequence as the new differences between consecutive terms. For example, since the fourth term in the GEB sequence is 12 and the fourth difference from the difference sequence is 6 , then the fifth term in the GEB sequence is $12+6=18$.

Continuing in this way, we can write out more terms in the GEB sequence:

$$
1,3,7,12,18,26,35,45,56, \ldots
$$

In a similar way, each positive integer less than 26 except 1, 3, 7, 12, 18 (a total of 20 positive integers) must occur in the difference sequence.

Since the difference sequence is increasing, then these 20 positive integers occur in increasing order.

Therefore, the difference sequence begins

$$
2,4,5,6,8,9,10,11,13,14,15,16,17,19,20,21,22,23,24,25, \ldots
$$

As above, we can write out more terms in the GEB sequence:

$$
1,3,7,12,18,26,35,45,56,69,83,98,114,131, \ldots
$$

Again, every positive integer less than 114, with the exception of the 12 integers before 114 in the GEB sequence, must occur in the difference sequence, and these integers $(113-12=101$ of them in all) must occur in increasing order.

We need to determine the 100th term in the GEB sequence. We can do this by taking the first term in the GEB sequence (that is, 1) and adding to it the first 99 terms in the difference sequence. This is because the terms in the difference sequence are the differences between consecutive terms in the GEB sequence, so adding these to the first term allows us to move along the sequence.

From above, we see that 113 is 101 st term in the difference sequence, so 112 is the 100th term, and 111 is the 99 th term.

Since the first 99 terms in the difference sequence consist of most of the integers from 2 to 111, with the exception of a few (those in the GEB sequence), we can find the sum of these terms by adding all of the integers from 2 to 111 and subtracting the relevant integers.

Therefore, the 100th term in the GEB sequence equals

$$
\begin{aligned}
& 1+(2+4+5+6+8+\cdots+109+110+111) \\
& =1+(1+2+3+4+\cdots+109+110+111) \\
& -(1+3+7+12+18+26+35+45+56+69+83+98) \\
& =1+\frac{1}{2}(111)(112)-(453) \\
& \text { (using the fact that the sum of the first } n \text { positive integers is } \frac{1}{2} n(n+1) \text { ) } \\
& =1+111(56)-453 \\
& =5764
\end{aligned}
$$

Thus, the 100th term in the GEB sequence is 5764 .