Prove that \( { }^{1} P_{1}+2 \cdot{ }^{2} P_{2}+3 \cdot{ }^{3} P_{3}+\cdots \cdot+n \cdot{ }^{n...

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Prove that \( { }^{1} P_{1}+2 \cdot{ }^{2} P_{2}+3 \cdot{ }^{3} P_{3}+\cdots \cdot+n \cdot{ }^{n} P_{n}={ }^{n+1} P_{n+1}-1 \)
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