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# binomial theorem

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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 $$\sum_{i = 0}^{2006} \frac{b_i}{c_i} - \sum_{i = 0}^{2005} \frac{a_i}{b_i}.$$

any help would be appreciated!

Feb 5, 2021