Induction/All quantifier/Exercise

Let ${}A(n)$ denote a statement (or rather a form of a statement) in which one can insert natural numbers. Discuss the difference between the two statements

${\left(\forall nA(n)\right)}\rightarrow {\left(\forall nA(n+1)\right)}\,\,{\text{ and }}\,\,\forall n{\left(A(n)\rightarrow A(n+1)\right)}.$

What is the mathematical relevance of the two statements?

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