jmc

Shown below are rows 1, 2, and 3 of Pascal's triangle.

$$\begin{array}{ccccccc}& & 1& & 1& & \\ & 1& & 2& & 1& \\ 1& & 3& & 3& & 1\end{array}$$Let $(a_i),$ $(b_i),$ $(c_i)$ be the sequence, from left to right, of elements in the 2005th, 2006th, and 2007th rows, respectively, with the leftmost element occurring at $i=0.$ Compute $$\sum _{i=0}^{2006}\frac{b_i}{c_i}-\sum _{i=0}^{2005}\frac{a_i}{b_i}.$$