Let two sequences $\left( u _ { n } \right)$ and $\left( v _ { n } \right)$ be defined by $u _ { 0 } = 2$ and $v _ { 0 } = 10$ and for every natural number $n$,

$$u _ { n + 1 } = \frac { 2 u _ { n } + v _ { n } } { 3 } \quad \text { and } \quad v _ { n + 1 } = \frac { u _ { n } + 3 v _ { n } } { 4 } .$$

\textbf{PART A}

Consider the following algorithm:

\begin{verbatim}
Variables: N is an integer
    U,V,W are real numbers
    K is an integer
Start: Assign 0 to K
    Assign 2 to U
    Assign 10 to V
    Input N
    While K<N
        Assign K+1 to K
        Assign U to W
        Assign (2U+V)/3 to U
        Assign (W+3V)/4 to V
    End while
    Display U
    Display V
End
\end{verbatim}

Execute this algorithm by inputting $N = 2$. Copy and complete the table given below showing the state of the variables during the execution of the algorithm.

\begin{center}
\begin{tabular}{ | c | c | c | c | }
\hline
$K$ & $W$ & $U$ & $V$ \\
\hline
0 &  &  &  \\
\hline
1 &  &  &  \\
\hline
2 &  &  &  \\
\hline
\end{tabular}
\end{center}

\textbf{PART B}

1. a. Show that for every natural number $n , v _ { n + 1 } - u _ { n + 1 } = \frac { 5 } { 12 } \left( v _ { n } - u _ { n } \right)$.

b. For every natural number $n$ let $w _ { n } = v _ { n } - u _ { n }$.

Show that for every natural number $n , w _ { n } = 8 \left( \frac { 5 } { 12 } \right) ^ { n }$.

2. a. Prove that the sequence $( u _ { n } )$ is increasing and that the sequence $( v _ { n } )$ is decreasing.

b. Deduce from the results of questions 1. b. and 2. a. that for every natural number $n$ we have $u _ { n } \leqslant 10$ and $v _ { n } \geqslant 2$.

c. Deduce that the sequences $\left( u _ { n } \right)$ and $\left( v _ { n } \right)$ are convergent.

3. Show that the sequences $( u _ { n } )$ and $( v _ { n } )$ have the same limit.

4. Show that the sequence $( t _ { n } )$ defined by $t _ { n } = 3 u _ { n } + 4 v _ { n }$ is constant.

Deduce that the common limit of the sequences $\left( u _ { n } \right)$ and $\left( v _ { n } \right)$ is $\frac { 46 } { 7 }$.