Prove that there exists no complex number z such that $$\left| z \right| < {1 \over 3}\,and\,\sum\limits_{r = 1}^n {{a_r}{z^r}} = 1$$ where $$\left| {{a_r}} \right| < 2$$.

If n^th divisions of the main scale coincide with (n+1)^th divisions of vernier scale. Given one main scale division is equal to 'a' units. Find the least count of the vernier.