New train stations

Let's n to be the number of stations before the new stations were created. The number of tickets before the new stations were created, $N_{b}$ , is

(1) $N_{b} = {{n}\choose{2}} = \frac{n(n-1)}{2}$

Let's a to be the additional number of stations created, so the final number of total tickets, $N_{a}$ , is

(2) $N_{a} = {{n + a}\choose{2}} = \frac{(n + a)(n + a - 1)}{2}$

Therefore

(3) $N_{a} - N_{b} = \frac{(n + a)(n + a - 1)}{2} - \frac{n(n - 1)}{2} = \frac{n^{2} + 2an + a^{2} - n - a - n^{2} + n}{2} = \frac{2na + a^{2} - a}{2} = \frac{a(2n + a - 1)}{2} = 17 \Rightarrow a(2n + a - 1) = 34$

Now we know that

1. a > 1

2. $34 = 2\times 17$ , which are prime factors.

3. 2n + a - 1 > a

Therefore

(4) a = 2

(5) 2n + a - 1 = 17

inserting (4) inside (5) we get $\boxed{n = 8}$