4,6,9,13,18,24

The given sequence is 4, 6, 9, 13, 18, 24. Observing the differences between consecutive terms:

The differences themselves form a sequence: 2, 3, 4, 5, 6, which increases by 1 each time. Following this pattern, the next difference is 7.

Next Term:

24 (last term) + 7 (next difference) = 31

Formula:

The sequence can be represented by the quadratic formula:

[

ext{Term}(n) = frac{n^2 + n + 6}{2} ]

For ( n = 7 ):

[

ext{Term}(7) = frac{7^2 + 7 + 6}{2} = frac{62}{2} = 31 ]

Answer: The next number in the sequence is 31.