B1. Determine the exact upper bound of the sequence with the general term $a_{n}=n\left(\frac{20}{23}\right)^{n}$ for all $n \in \mathbb{N}$.
B1. Let's look at when the sequence increases. The inequality $a_{n+1}>a_{n}$, considering the formula for the nth term of the sequence, becomes $(n+1)\left(\frac{20}{23}\right)^{n+1}>n\left(\frac{20}{23}\right)^{n}$ and is equivalent to the inequality $(n+1) \cdot \frac{20}{23}>n$. Simplifying the inequality gives $20 n+20>23 n$, from which it follows that $n<\frac{20}{3}$. The inequality $a_{n+1}>a_{n}$ is therefore satisfied for $n \in\{1,2,3,4,5,6\}$. This means that the sequence $a_{n}$ increases up to the term $a_{7}$, and then starts to decrease. Its exact upper bound is thus equal to the value $a_{7}=7\left(\frac{20}{23}\right)^{7}$.

Process 1:

a) determination that the sequence increases up to a certain value, then decreases ..................... 2 points

![](https://cdn.mathpix.com/cropped/2024_06_07_95af3fbbcea2fb458cf3g-16.jpg?height=49&width=1727&top_left_y=2237&top_left_x=176)

![](https://cdn.mathpix.com/cropped/2024_06_07_95af3fbbcea2fb458cf3g-16.jpg?height=63&width=1727&top_left_y=2284&top_left_x=176)

![](https://cdn.mathpix.com/cropped/2024_06_07_95af3fbbcea2fb458cf3g-16.jpg?height=57&width=1725&top_left_y=2339&top_left_x=177)

d) calculation of the maximum value $a_{7}=7\left(\frac{20}{23}\right)^{7} \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots$

Comment: The competitor does not lose points at c) due to a calculation error at b2).

The competitor can only get the point d) if they get the point c).

The competitor who correctly solves part b) automatically gets the points for part a).

Process 2:

a1) The extremum can be at three values: $\mathrm{n}=1$, at the point $n, n+1$, so that $f^{\prime}(x)$ is somewhere in the interval $(n, n+1)$, at infinity .1 point

a2) Exclude the possibilities $n=1$ and infinity .1 point

b1) write the equality for the derivative: $a(x)^{\prime}=0$........................................................... 1 point

![](https://cdn.mathpix.com/cropped/2024_06_07_95af3fbbcea2fb458cf3g-17.jpg?height=68&width=1739&top_left_y=480&top_left_x=176)

![](https://cdn.mathpix.com/cropped/2024_06_07_95af3fbbcea2fb458cf3g-17.jpg?height=55&width=1739&top_left_y=538&top_left_x=176)

d) calculation of the maximum value $a_{7}=7\left(\frac{20}{23}\right)^{7} \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots$ point

Comment: The competitor can only get the point d) if they get the point c).