Problem 1. Prove each statement using induction. (a) For all ( n in mathbb{N}, 1+2+2^{2}+2^{3}+2^{4}+ cdots+2^{n}=2^{n+1}-1 ). (b) For all ( n in mathbb{N}, 4^{n}-1 ) is divisible by 3 . (c) For all ( n in mathbb{N} ) with ( n geq 2,2^{n}n+1 ).

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Problem 1. Prove each statement using induction. (a) For all  ( n  in  mathbb{N}, 1+2+2^{2}+2^{3}+2^{4}+ cdots+2^{n}=2^{n+1}-1  ). (b) For all  ( n  in  mathbb{N}, 4^{n}-1  ) is divisible by 3 . (c) For all  ( n  in  mathbb{N}  ) with  ( n  geq 2,2^{n}>n+1  ).

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Solution for -   Problem 1. Prove each statement using induction. (a) For all  ( n  in  mathbb{N}, 1+2+2^{2}+2^{3}+

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(Solved): Problem 1. Prove each statement using induction. (a) For all \( n \in \mathbb{N}, 1+2+2^{2}+2^{3}+ ...

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