Main topics covered: Numerical sequences; proof by induction; geometric sequences. The sequence $(u_{n})$ is defined on $\mathbb{N}$ by $u_{0} = 1$ and for every natural number $n$, $$u_{n+1} = \frac{3}{4}u_{n} + \frac{1}{4}n + 1.$$
Calculate, showing the calculations in detail, $u_{1}$ and $u_{2}$ in the form of irreducible fractions.
The extract, reproduced below, from a spreadsheet created with a spreadsheet application presents the values of the first terms of the sequence $(u_{n})$.
A
B
1
$n$
$u_{n}$
2
0
1
3
1
1.75
4
2
2.5625
5
3
3.421875
6
4
4.31640625
a. What formula, then extended downward, can be written in cell B3 of the spreadsheet to obtain the successive terms of $(u_{n})$ in column B? b. Conjecture the direction of variation of the sequence $(u_{n})$.
a. Prove by induction that, for every natural number $n$, we have: $n \leqslant u_{n} \leqslant n+1$. b. Deduce from this, justifying the answer, the direction of variation and the limit of the sequence $(u_{n})$. c. Prove that: $$\lim_{n \rightarrow +\infty} \frac{u_{n}}{n} = 1$$
We denote by $(v_{n})$ the sequence defined on $\mathbb{N}$ by $v_{n} = u_{n} - n$ a. Prove that the sequence $(v_{n})$ is geometric with common ratio $\frac{3}{4}$. b. Deduce from this that, for every natural number $n$, we have: $u_{n} = \left(\frac{3}{4}\right)^{n} + n$.
\textbf{Main topics covered: Numerical sequences; proof by induction; geometric sequences.}
The sequence $(u_{n})$ is defined on $\mathbb{N}$ by $u_{0} = 1$ and for every natural number $n$,
$$u_{n+1} = \frac{3}{4}u_{n} + \frac{1}{4}n + 1.$$
\begin{enumerate}
\item Calculate, showing the calculations in detail, $u_{1}$ and $u_{2}$ in the form of irreducible fractions.
\end{enumerate}
The extract, reproduced below, from a spreadsheet created with a spreadsheet application presents the values of the first terms of the sequence $(u_{n})$.
\begin{center}
\begin{tabular}{|c|c|c|}
\hline
& A & B \\
\hline
1 & $n$ & $u_{n}$ \\
\hline
2 & 0 & 1 \\
\hline
3 & 1 & 1.75 \\
\hline
4 & 2 & 2.5625 \\
\hline
5 & 3 & 3.421875 \\
\hline
6 & 4 & 4.31640625 \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}
\setcounter{enumi}{1}
\item a. What formula, then extended downward, can be written in cell B3 of the spreadsheet to obtain the successive terms of $(u_{n})$ in column B?\\
b. Conjecture the direction of variation of the sequence $(u_{n})$.
\item a. Prove by induction that, for every natural number $n$, we have: $n \leqslant u_{n} \leqslant n+1$.\\
b. Deduce from this, justifying the answer, the direction of variation and the limit of the sequence $(u_{n})$.\\
c. Prove that:
$$\lim_{n \rightarrow +\infty} \frac{u_{n}}{n} = 1$$
\item We denote by $(v_{n})$ the sequence defined on $\mathbb{N}$ by $v_{n} = u_{n} - n$\\
a. Prove that the sequence $(v_{n})$ is geometric with common ratio $\frac{3}{4}$.\\
b. Deduce from this that, for every natural number $n$, we have: $u_{n} = \left(\frac{3}{4}\right)^{n} + n$.
\end{enumerate}