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Question Number 98880 by Ar Brandon last updated on 16/Jun/20

$$\mathcal{G}\mathrm{iven}\mathrm{the}\mathrm{sequence}{\left({\mathrm{u}}_{\mathrm{n}}\right)}_{\mathrm{n}\in {\mathbb{N}}^{\ast }}\mathrm{defined}\mathrm{by}\{\begin{array}{l}\frac{1}{2^{\mathrm{n}}}\mathrm{if}\mathrm{n}\equiv 0\mathrm{mod}\left(3\right)\\ \frac{1}{3^{\mathrm{n}}}+1\mathrm{if}\mathrm{n}\equiv 1\mathrm{mod}\left(3\right)\\ \frac{{\mathrm{u}}_{\mathrm{n}-1}+{\mathrm{u}}_{\mathrm{n}+2}}{2}\mathrm{if}\mathrm{n}\equiv 2\mathrm{mod}\left(3\right)\end{array}$$$$\mathrm{a}\mathrm{\setminus }\mathcal{D}\mathrm{etermine}\mathrm{the}\mathrm{first}-8^{\mathrm{th}}\mathrm{terms}\mathrm{of}{\left({\mathrm{u}}_{\mathrm{n}}\right)}_{\mathrm{n}\in {\mathbb{N}}^{\ast }}$$$$\mathrm{b}\mathrm{\setminus }\mathcal{S}\mathrm{how}\mathrm{that}\mathrm{the}\mathrm{sequences}{\left({\mathrm{v}}_{\mathrm{n}}\right)}_{\mathrm{n}\in \mathbb{N}},{\left({\mathrm{w}}_{\mathrm{n}}\right)}_{\mathrm{n}\in \mathbb{N}},\mathrm{and}{\left({\mathrm{z}}_{\mathrm{n}}\right)}_{\mathrm{n}\in \mathbb{N}}$$$$\mathrm{where}{\mathrm{v}}_{\mathrm{n}}={\mathrm{u}}_{3\mathrm{n}},{\mathrm{w}}_{\mathrm{n}}={\mathrm{u}}_{3\mathrm{n}+1}\mathrm{and}{\mathrm{z}}_{\mathrm{n}}={\mathrm{u}}_{3\mathrm{n}+2}\mathrm{are}\mathrm{convervent}$$$$\mathrm{and}\mathrm{find}\mathrm{their}\mathrm{respective}\mathrm{limits}$$$$\mathrm{c}\mathrm{\setminus }\mathcal{D}\mathrm{educe}\mathrm{the}\mathrm{nature}\mathrm{of}{\left({\mathrm{u}}_{\mathrm{n}}\right)}_{\mathrm{n}\in {\mathbb{N}}^{\ast }}$$

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