Showing that some term of a sequence is 0 I have a sequence defined thus : I start with $x_1 = n$. Then $x_{k}$ is defined by first writing $x_{k-1}$ in terms of base $k$ expansion, i.e. $$x_{k-1} = a_o+a_1k+a_2k^2+\cdots$$ and then reading it in base $k+1$ and finally subtracting 1 i.e. $$x_k = a_0 +a_1(k+1) +a_2(k+1)^2 +\cdots -1.$$ I need to show that I can find an $N$ for which $x_N=0$.
I am interested in hints or suggestions for approaching the problem and not the solution.
I checked out the sequence for some values of $n =1,2,3,4,5$ and see that the trend before 0 is hit is the appearance of consecutive numbers $3,2,1$. For $n =4$, the sequence seems to be $4,8,7,7,\cdots,6,5,\cdots,0$ and for $n=5$, it reads $5,9,15,17,19,21,23,24,25,26,25,24,23,\cdots, 22,\cdots 0$.
So the sequence increases and slowly peters out decreasing to hit 0.  I see that eventually $k$ soon overtakes the $x_k$ value at some point and that is when the decreasing trend is seen. But I am not sure how this helps.
 A: I've changed my mind. Considering $n$ to be non-negative (it doesn't work otherwise!), 
this viewpoint might be helpful.
Collecting the coefficients ($a_j$'s), you can naturally represent each point in the sequence $(x_k)_{k \geq 2}$ as an integer tuple in $\mathbb Z^{N+1}$, where $N$ is the largest power of 2 which appears in the binary expansion of $n$.
$$x_k = \sum_{j=0}^N a_jk^j \quad\longleftrightarrow\quad y_k = (a_0,\ldots, a_N),$$
where the $a_j$ are all understood to be non-negative. The length doesn't need to change, so you can deduce what you need from the corresponding sequence $(y_k)_{k = 2}^\infty \subset \mathbb Z^{N+1}$.
In particular, the corresponding rule for obtaining $y_{k+1}$ from $y_k$ is quite simple (I invite you to think about it):

 $$ y_k = (a_0, a_1, \ldots, a_N) \longmapsto y_{k+1} = (a_0 - 1, a_1, \ldots, a_N),$$
 
 where, whenever you see a $-1$ in any entry, you should replace this with a
 $k$ and subtract one from the next entry along: e.g. if $k = 9$, $(-1,1) = (9,0)$
 which corresponds to $10 - 1 = 9$, so that the $a_k$'s are all non-negative.

and so once you've figured that out/peeked, it should be hopefully much clearer why you at some point obtain $x_k = 0$ in your sequence—a.k.a. $y_k = (0,0,\ldots, 0)$.
