THE ANSWER:
In each case, the answer is FOUR. The next number in each sequence is the number of letters used to spelling out the previous number.
SEVENTEEN (9 letters) -> NINE (4 letters) -> FOUR
ONE (3 letters) -> THREE (5 letters) -> FIVE (4 letters) -> FOUR
TWENTY (6 letters) -> SIX (3 letters) -> THREE (5 letters) -> FIVE (4 letters) -> FOUR
They all get stuck on the number four, since its name has four letters.
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