Think Bases

The title of the problem gives us a hint about different bases. Let's treat each number in base $2$ and convert them in base $10$

\[\begin{align} 1_{2} &\implies 1 \\ 11_{2} &\implies 3 \\ 110_{2} &\implies 6 \\ 1010_{2} &\implies 10 \\ 1111_{2} &\implies 15 \\ 10101_{2} &\implies 21

\end{align}\]

The pattern we see is that the $i^{\text{th}}$ term in this sequence when converted from base $2$ to base $10$ is the $i^{\text{th}}$ triangle number . According to this rule, the next term should be $28$ in base $10$ . Which is:

$28 = \boxed{11100_{2}}$

If you convert each number to base 2, you get the following sequence:

1,3,6,10,15,21,?

This is a very simple sequence that works like this:

x, x+1, (x+1)+2, ([x+1]+2)+3, ([{x+1}+2]+3)+4, ...

So the next term in the sequence is 21+7=28. Converting 28 to base 2 we get $\boxed{11100}$ .