I want to know if my calculations are correct. I am trying to find the fundamental group of the figure $X_n$ in $\mathbb{R}^2$ obtained by drawing $n$ lines between two points (the lines only meet at those two points).
First two cases are trivial and I found that for $n=3$, using Seifert-Van Kampen's theorem, is$$\pi_1(X_3)=Z\langle a_1,a_2\rangle$$ so my guess would be that $$\pi_1(X_n)=Z\langle a_1,\ldots,a_{n-1}\rangle$$ is this correct? and if so, would induction work?. I don't manage to do it that way.