Given-the-sequence-u-n-n-N-defined-by-1-2-n-if-n-0mod-3-1-3-n-1-if-n-1mod-3-u-n-1-u-n-2-2-if-n-2mod-3-a-Determine-the-first-8-th-terms-of-u-n-n-N

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

G iven the sequence {(u_{n})}_{n \in N^{*}} defined by {\begin{cases} \frac{1}{2^{n}} if n \equiv 0 \mod (3) \\ \frac{1}{3^{n}} + 1 if n \equiv 1 \mod (3) \\ \frac{u_{n - 1} + u_{n + 2}}{2} if n \equiv 2 \mod (3) \end{cases}

a ∖ D etermine the first - 8^{th} terms of {(u_{n})}_{n \in N^{*}}

b ∖ S how that the sequences {(v_{n})}_{n \in N}, {(w_{n})}_{n \in N}, and {(z_{n})}_{n \in N}

where v_{n} = u_{3 n}, w_{n} = u_{3 n + 1} and z_{n} = u_{3 n + 2} are convervent

and find their respective limits

c ∖ D educe the nature of {(u_{n})}_{n \in N^{*}}