Find the missing term

Question

The sequence $10000, 121, 100, 31, 24, n, 20$ represents a number $x$ with respect to different bases. What is the missing number, $n$?

This is from my elementary computer aptitude paper. Is there any way to solve this quickly?

Answer 1

If the base in the last term is $b$, the number is $2b$. The missing term is then in base $b-1$. Expressed in base $b-1$ the integer is $2(b-1)+2=22$ The third to last term shows that $b-2 \ge 5$ so we are safe from a carry.

Answer 2

The bases are 2,3,4,5,6,7,8, expressing values 1*16=16, 9 + 6 + 1 = 16, 1*16=16, 15 + 1 = 16, 12 + 4 = 16, 7*x + y = 16, and 2*8 = 16. x and y are then 2 and 2.

Answer 3

Assuming the bases are in ascending order with a difference of 1, and the first one is in base $b$, then $$b^4=(b+2)^2$$ and so $$b^2=b+2$$ from which we can conclude that $b=2$ and the number is 16, which in base 7 is represented as 22.