The number sequence $1,4,5,7,12,15,16,18,23, \ldots$ was written based on a certain rule. Continue the number sequence until it exceeds the number 50.

Continue the number sequence until it exceeds the number 50.
Let's calculate how much larger each term is compared to the one preceding it (starting from the second term). The increments are

$$
3,1,2,5,3,1,2,5
$$

We can see that starting from the 5th term, the first four increments repeat in the original order. Therefore, we can assume that this is a certain rule. Continuing the sequence by increasing each term by 3, 1, 2, and 5 respectively, up to the 50th number:

$$
26,27,29,34,37,38,40,45,48,49,51
$$

András Letenoczki (Budapest, II. district, Lajos u. 34. sz. elementary school VII. grade)

Note. The repetition of the increments every four terms can also be utilized by noting that starting from the 5th term, each term is $3+1+2+5=11$ larger than the term four places before it. This is more evident from the following arrangement of the sequence:

| 1, | 12, | 23, | 34, | 45, |
| :---: | :---: | :---: | :---: | :---: | :---: |
| 4, | 15, | 26, | 37, | 48, |
| 5, | 16, | 27, | 38, | 49, |
| 7, | 18, | 29, | 40, | 51. |

Kornélia Szász (Budapest, Szilágyi E. secondary school I. grade)