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Using the principle of mathematical induction prove that for all \( n \in N \) :
\[
\left(1-\frac{1}{2^{2}}\right)\left(1-\frac{1}{3^{2}}\right)\left(1-\frac{1}{4^{2}}\right) \cdots\left(1-\frac{1}{(n+1)^{2}}\right)=\frac{n+2}{2(n+1)} \text {. }
\]
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