Limit of $b_n$ when $b_n=\frac{a_n}{n}$ and $\lim\limits_{n\to\infty}(a_{n+1}-a_n)=l$ [duplicate]
As @ParamanandSingh suggested, from Stolz–Cesàro theorem $$a_{n+1}-a_n=\frac{a_{n+1}-a_n}{(n+1)-n} \rightarrow l, n \rightarrow \infty$$ where $\{n\}_{n \in \mathbb{N}}$ is monotone and divergent, then $$\frac{a_n}{n} \rightarrow l, n \rightarrow \infty$$