The Missing Term

A student was copying out a sequence for his homework when the telephone rang. While he was gone, his younger brother rubbed out one of the terms. Here is what was left: $$10,\ 11,\ 12,\ 13,\ 14,\ 15,\ 16,\ 17,\ 20,\ 22,\ 24,\ \rule{1.5em}{0.4pt}\,,\ 100,\ 121,\ 10000.$$ What was the number that was rubbed out?

Solution

Every term is the number sixteen, written in a different base, with the bases counting down from sixteen to two.

In base sixteen, sixteen is written $10$; in base fifteen, $11$; in base fourteen, $12$; and so on down to base ten, where it is the familiar $16$, then base nine ($17$), and base eight ($20$, since $16 = 2 \times 8$), base seven ($22$), base six ($24$). The next base down is five, and $16 = 3 \times 5 + 1$, so there sixteen is written $31$. The list then finishes with base four ($100$, since $16 = 4^2$), base three ($121$, since $16 = 9 + 6 + 1$) and base two ($10000$).

The missing term is $31$, which is sixteen written in base five.