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A connected graph \( G(V, E) \) is said to be \((a, d)\)-antimagic if there exist positive integers \( a \) and \( d \) and a bijection \( f: E \to \{1, 2, \ldots, |E|\} \) such that the induced mapping \( \text{g}_\text{f}: V \to \mathbb{N} \) defined by \( \text{g}_\text{f}(v) = \sum_{\text{e} \in \text{I}(v)} \text{f(e)} \), where \( \text{I}(v) = \{\text{e} \in E \mid \text{e} \text{ is incident to } v\} \), \( v \in V \) is injective and \( \text{g}_\text{f}(V) = \{a, a+d, a+2d, \ldots, a+(|V|-1)d\} \). In this paper, using partition, we prove that (i) the 1-sided infinite path \( P_1 \) is \((1, 2)\)-antimagic, (ii) the path \( P_{2n+1} \) is \((n, 1)\)-antimagic, and (iii) the \((n+2, 1)\)-antimagic labeling is the unique \((a, d)\)-antimagic labeling of \( C_{2n+1} \); and the graphs \( K_1 + (K_1 \cup K_2) \), \( P_{2n} \), and \( C_{2n} \) are not \((a, d)\)-antimagic. For \( a, d \in \mathbb{N} \), on an \((a, d)\)-antimagic graph \( G \), we obtain a new relation, \( a + (p-1)d \leq \frac{\Delta(2q – \Delta + 1)}{2} \). Using the results on \((a, d)\)-antimagic labeling of \( C_{2n} \) and \( C_{2n+1} \), we obtain results on the existence of \((a, d)\)-arithmetic sequences of length \( 2n \) and \( 2n+1 \), respectively.