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

Consider the cross-section of the roll.
It might be easier to think of it as a continuous spool of paper that's later going to be divided lengthwise equally into sheets.
The circumference of the cross section at a radial distance $r$ from the centre is $2\pi r$. The thickness of one layer is $t$. The circumference one layer outward will be $2\pi (r+t) $, which is $2\pi t$ more. All this is not necessary, it's just to show that adding a layer adds a constant to the cross sectional circumference.
If we now think in terms of sheets, we can say the initial cross-sectional circumference is $1$ sheet while the final is $2$ sheets. You're adding a constant each time. This is an arithmetic progression. 
The sum of an arithmetic series can be given by different formulas. The easiest one to use here is $S(n) = \frac n2 (a+l) $, where $n$ is the number of terms (equal to number of layers, and this is what you need to solve for). $a$ is the first term ($1$ here) and $l$ is the final term ($2$ here). You can also think of it like $n$ times the average term, which allows you to relate it to your given solution.
So $\frac n2 (1+2) = 180 \implies n =120$.
