## Binomial Theorem Corollaries

Here are some simple proofs based on the binomial theorem. They also demonstrate how amazing mathematical proofs can be. 1. The hockey stick identity By the formula for the sum of a geometric series, \(\frac{(1+r)^{n+1}-1}{(1+r)-1}=\displaystyle\sum_{i=0}^n (1+r)^i\). Now expanding both sides by the binomial theorem and simplifying, $$\displaystyle\sum_{i=0}^n \binom{n+1}{i+1}r^i=\displaystyle\sum_{i=0}^n \displaystyle\sum_{j=0}^i \binom{i}{j}r^j$$ Now rearrange the sum on […]

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