Stick's triangles

The number of triangles increases by the previous increase plus one unit, and becomes the triangular number . Since each triangle is built with three sticks, the formation pattern is given by:

$S(n)= \frac {3.n.(n+1)}{2}$ .

Then, we have:

$135= \frac {3.n.(n+1)}{2} \rightarrow n^{2}+n-90=0 \rightarrow \boxed{n=9}$ .

Note : the other root of the polynomial is negative and, thus, discarded.

We can representate the number of sticks of each triangle as $3 \times n + m$ where $n$ is the number of stick in each side and $m$ is the number of sticks in the subsequent triangle.

Applying the formula we have the following number of sticks in each triangle:

$3, 9, 18, 30, 45, 63, 84, 108, 135...$

Then:

$3n + 108 = 135$

$3n =27$

$n =9$

The answer is $\boxed {9}$