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Using the principle of mathematical induction prove that for all \( n \in N \) :
\[
\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\ldots+\frac{1}{2^{n}}=1-\frac{1}{2^{n}} .
\]
\( \mathrm{P} \)
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