The outside of a $180$-sheet roll of toilet paper is covered by two sheets; the inner cylinder, by one. What's wrong with how I counted the layers? 
Puzzle: A roll of toilet paper has 180 sheets on it. The outside is covered with exactly two sheets. The inside around the cardboard cylinder is covered by exactly one. Question of the puzzle: how many layers of toilet paper are on the roll of toilet paper?

The given solution: One way to solve this is by saying that the average round is covered by 1.5 sheets, so therefore the answer is $180\times\frac{2}{3}=120$
I tried a similar (but wrong) reasoning: "the average sheet makes an average of $\frac{3}{4}$ rounds (first sheet makes one round and the last sheet makes $\frac{1}{2}$ rounds), so the answer is $180\times\frac{3}{4}=$ 135"

QUESTION: Apparently my answer is wrong. But since it seems analogical to the given solution I don't understand what error I made.

Possibly the growth of sheets per round is constant? While the (negative) growth of rounds per sheet is not constant? What are the related functions?
Put in another way: if $\frac{dSheets}{dRounds}=Constant$ isn't also $\frac{dRounds}{dSheets}=Constant$?
This question is linked to this question: Using differential equations to determine the number of rolls on a roll of toilet paper

To solve more rigorously, call $L$ the length of a single sheet, $r$ and $R$ the inner and outer radius of the roll, respectively and $n$ the number of layers.
From the information given, we know that $R = 2\pi L = 2r$, and also
$$
\begin{cases}
(n-1)\Delta R = R - r = r\\
180 L = 2\pi\sum_{k=0}^{n-1} R_k,
\end{cases}
$$
where $\Delta R$ is the reduction in radius after a layer and $ R_k $ is the radius of the roll after $k$ layers, which we can assume to be constant on that layer provided $\Delta R \ll \min_k R_k = r$ (which we can check at the end).
Then $ R_k = R-k\Delta R $ and the sum gives us
$$
180 L = \pi n (R+r),
$$
and applying all the identities between these quantities this simplifies to
$$
n = 120.
$$
As a final consistency check, notice that $ \Delta R = r/(n-1) = r/119 \ll r $ as required.
