Given:
13,18,16,21,19

We note:

18=13+5 and thus the second term is the first term increased by 5

16=18-2 and thus the third term is the second term decreased by 2

21=16+5 and thus the fourth term is the third term increased by 5

19=21-2 and thus the fifth term is the fourth term decreased by 2

Thus we note that we alternately increase the previous term by 5 and decreased the previous term by 2.

We last decreased by 2 and thus we expect the next term to be the last term increased by 5.

19+5=24

Thus we expect the next number in the series to be 24.

We note:

18=13+5 and thus the second term is the first term increased by 5

16=18-2 and thus the third term is the second term decreased by 2

21=16+5 and thus the fourth term is the third term increased by 5

19=21-2 and thus the fifth term is the fourth term decreased by 2

Thus we note that we alternately increase the previous term by 5 and decreased the previous term by 2.

We last decreased by 2 and thus we expect the next term to be the last term increased by 5.

19+5=24

Thus we expect the next number in the series to be 24.