We consider the sequence $( u _ { n } )$ defined by $u _ { 0 } = 3$ and, for every natural number $n$,
$$u _ { n + 1 } = \frac { 1 } { 2 } u _ { n } + \frac { 1 } { 2 } n + 1 .$$
Part A
This part is a multiple choice questionnaire. For each of the following questions, only one of the four proposed answers is correct. To answer, indicate on your paper the question number and the letter of the chosen answer. No justification is required.
A wrong answer, no answer, or multiple answers, neither earn nor lose points.
- The value of $u _ { 2 }$ is equal to: a. $\frac { 11 } { 4 }$ b. $\frac { 13 } { 2 }$ b. 2.7 c. 3.5
- The sequence $\left( v _ { n } \right)$ defined, for every natural number $n$, by $v _ { n } = u _ { n } - n$ is: a. arithmetic with common difference $\frac { 1 } { 2 }$ b. geometric with common ratio $\frac { 1 } { 2 }$ c. constant. d. neither arithmetic nor geometric.
- We consider the function below, written incompletely in Python language. $n$ denotes a non-zero natural number. We recall that in Python language ``i in range (n)'' means that i varies from 0 to $n - 1$.
| 1 | def terme $( \mathrm { n } )$ |
| 2 | $\mathrm { U } = 3$ |
| 3 | for i in range(n) : |
| 4 | $\ldots \ldots \ldots \ldots \ldots \ldots \ldots$ |
| 5 | return U |
For terme(n) to return the value of $u _ { n }$, we can complete line 4 by: a. $\mathrm { U } = \mathrm { U } / 2 + ( \mathrm { i } + 1 ) / 2 + 1$ b. $\mathrm { U } = \mathrm { U } / 2 + \mathrm { n } / 2 + 1$ c. $U = U / 2 + ( i - 1 ) / 2 + 1$ d. $\mathrm { U } = \mathrm { U } / 2 + \mathrm { i } / 2 + 1$
Part B
- Prove by induction that for every natural number $n$: $$n \leqslant u _ { n } \leqslant n + 3 .$$
- Deduce the limit of the sequence $( u _ { n } )$.
- Determine the limit of the sequence $\left( \frac { u _ { n } } { n } \right)$.