Identifying the Pattern
To solve the sequence: 3, 4, 7, 7, 13, 13, 21, 22, 31, 34, we first need to analyze the differences between consecutive numbers to uncover any underlying pattern.

Calculating Differences
- 4 - 3 = 1
- 7 - 4 = 3
- 7 - 7 = 0
- 13 - 7 = 6
- 13 - 13 = 0
- 21 - 13 = 8
- 22 - 21 = 1
- 31 - 22 = 9
- 34 - 31 = 3
The differences between the terms are: 1, 3, 0, 6, 0, 8, 1, 9, 3.

Finding a Pattern in Differences
Now, let's look at the differences again to see if they follow any pattern:
- 1, 3, 0, 6, 0, 8, 1, 9, 3
We can group the differences:
- 1, 3
- 0, 6
- 0, 8
- 1, 9
- 3
It seems there is an alternating pattern with some repetitions.

Predicting the Next Number
To predict the next number in the sequence, consider the last few differences:
- The last difference was 3, and before that, it was 9.
A logical approach would be to add the next difference which follows a pattern (seems to alternate between smaller values and larger jumps).
Following this pattern, the next difference might likely be 7 (following the increasing trend).
Thus, we can calculate:
34 + 7 = 41

Conclusion
The next number in the sequence is likely **41**.