Main topics covered: Numerical sequences; proof by induction.

We consider the sequences $(u_n)$ and $(v_n)$ defined by:

$$u_0 = 16 \quad; \quad v_0 = 5$$

and for any natural number $n$:

$$\left\{\begin{aligned}
u_{n+1} & = \frac{3u_n + 2v_n}{5} \\
v_{n+1} & = \frac{u_n + v_n}{2}
\end{aligned}\right.$$

\begin{enumerate}
  \item Calculate $u_1$ and $v_1$.
  \item We consider the sequence $(w_n)$ defined for any natural number $n$ by: $w_n = u_n - v_n$.\\
a. Prove that the sequence $(w_n)$ is geometric with ratio 0.1.

Deduce from this, for any natural number $n$, the expression of $w_n$ as a function of $n$.\\
b. Specify the sign of the sequence $(w_n)$ and the limit of this sequence.
  \item a. Prove that, for any natural number $n$, we have: $u_{n+1} - u_n = -0.4w_n$.\\
b. Deduce that the sequence $(u_n)$ is decreasing.

It can be shown in the same way that the sequence $(v_n)$ is increasing. We admit this result, and we note that we then have: for any natural number $n$, $v_n \geq v_0 = 5$.\\
c. Prove by induction that, for any natural number $n$, we have: $u_n \geq 5$.

Deduce that the sequence $(u_n)$ is convergent. We call $\ell$ the limit of $(u_n)$.\\
It can be shown in the same way that the sequence $(v_n)$ is convergent. We admit this result, and we call $\ell'$ the limit of $(v_n)$.
  \item a. Prove that $\ell = \ell'$.\\
b. We consider the sequence $(c_n)$ defined for any natural number $n$ by: $c_n = 5u_n + 4v_n$. Prove that the sequence $(c_n)$ is constant, that is, for any natural number $n$, we have: $c_{n+1} = c_n$.\\
Deduce that, for any natural number $n$, $c_n = 100$.\\
c. Determine the common value of the limits $\ell$ and $\ell'$.
\end{enumerate}