360
The assumption implies that there holds $ia_i = a_1 + (i-1)(2a_2 - a_1)$ for any integer $2 \le i \le 7$. Subtracting $a_1$ from both sides and dividing by $i$, one obtains
$$a_i - a_1 = \frac{ia_i - ia_1}{i} = \frac{(i-1)(2a_2 - a_1) - (i-1)a_1}{i} = \frac{2(i-1)(a_2 - a_1)}{i}.$$
Since $i$ and $i-1$ are coprime, $2(a_2 - a_1)$ is divided by $i$ for any $2 \le i \le 7$, thus divided by $420$. By the assumption $a_2 - a_1 \ne 0$, thus $|2(a_2 - a_1)| \ge 420$. Hence
$$|a_7 - a_1| = \left| \frac{2 \cdot 6 \cdot (a_2 - a_1)}{7} \right| \ge \frac{6}{7} \cdot 420 = 360.$$
On the other hand, the sequence $(a_1, a_2, a_3, a_4, a_5, a_6, a_7) = (420, 210, 140, 105, 84, 70, 60)$ satisfies the assumption and $|a_7 - a_1| = 360$, thus the answer is $360$.