A new study on the strongly lacunary quasi Cauchyness

In this paper, the concept of an $N_{\theta }^2$ quasi-Cauchy sequence is introduced. We proved interesting theorems related to $N_{\theta }^2$-quasi-Cauchy sequences. A real valued function $f$ defined on a subset $A$ of $\mathbb{R}$, the set of real numbers, is $N_{\theta }^2$ ward continuous on $A$ if it preserves $N_{\theta }^2$ quasi-Cauchy sequences of points in $A$, i.e. $(f({\alpha }_k))$ is an $N_{\theta }^2$ quasi-Cauchy sequence whenever $({\alpha }_k)$ is an $N_{\theta }^2$ quasi-Cauchy sequences of points in $A$, where a sequence $({\alpha }_k)$ is called $N_{\theta }^2$ quasi-Cauchy if $({\mathrm{\Delta }}^2{\alpha }_k)$ is an $N_{\theta }$ quasi-Cauchy sequence where ${\mathrm{\Delta }}^2{\alpha }_k={\alpha }_{k+2}-2{\alpha }_{k+1}+{\alpha }_k$ for each positive integer $k$.