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Prove the telescoping property of products

Use induction to prove that if a_k \neq 0 for all k = 0, \ldots, n, then

    \[ \prod_{k=1}^n \frac{a_k}{a_{k-1}} = \frac{a_n}{a_0}. \]

This is the telescoping property for products.


Proof. For the case n=1 we have,

    \[ \prod_{k=1}^1 \frac{a_k}{a_{k-1}} = \frac{a_1}{a_0}. \]

Thus, the property holds for the case n=1. Assume then that it holds for some n = m \in \mathbb{Z}_{>0}. Then,

    \[ \prod_{k=1}^{m+1} \frac{a_k}{a_{k-1}} = \left( \frac{a_{m+1}}{a_m} \right) \cdot \prod_{k=1}^m \frac{a_k}{a_{k-1}} = \left( \frac{a_{m+1}}{a_m} \right) \left( \frac{a_m}{a_0} \right) = \frac{a_{m+1}}{a_0}. \]

Hence the property is true for m+1; and thus, for all n \in \mathbb{Z}_{>0}. \qquad \blacksquare

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