10, 11, 12, 13, 14, 20, 22, 101, what next?

Question

Find the rule and the next term of the sequence

10, 11, 12, 13, 14, 20, 22, 101, ?

The jump from 14 to 20 and then to 101 is the clue. Explain why the terms grow slowly at first and then suddenly acquire extra digits, and say whether the sequence has a term after the one you find.

Accepted Answer

The next term is 1010. Every term is the number ten, written in bases 10, 9, 8, 7, 6, 5, 4, 3 and finally 2. Check. Ten in base 9 is one nine and one unit: 11. In base 8: one eight and two units: 12. Base 7: 13. Base 6: 14. Base 5: two fives: 20. Base 4: two fours and two units: 22. Base 3: one nine, no threes, one unit: 101. Base 2: eight plus two: 1010. Why the jumps. As the base falls below half of ten, a single "digit" can no longer hold a big chunk of the number, so the representation needs more digits and grows quickly: 14 to 20 to 22 to 101 to 1010. There is no base-1 term in the usual…

10, 11, 12, 13, 14, 20, 22, 101, what next?

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