The exercise defines a sequence $(u_n)$ by:

$u_0 = 1, \quad u_{n+1} = 1.1 \cdot u_n + 3$

Let's break down the tasks provided in the image:

1. Write an algorithm to compute the value of $u_n$ given a number $n$

This requires creating an algorithm that starts with $u_0 = 1$ and calculates $u_n$ iteratively using the recurrence relation $u_{n+1} = 1.1 \cdot u_n + 3$.

Pseudocode:

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def calculate_u_n(n):
    u = 1  # Initial value u_0
    for i in range(1, n + 1):
        u = 1.1 * u + 3
    return u

2. Implement the algorithm to calculate $u_{10}, u_{100}, u_{500}, u_{1000}$. What conjecture can be made about the limit of the sequence $(u_n)$?

By using the algorithm from step 1, we calculate these specific values. As $n$ grows large, we expect the sequence to stabilize due to the increasing multiplier $1.1$. To explore the behavior, we can analyze the growth trend.

3. Write an algorithm to compute the smallest $n$ such that $u_n > M$, given a number $M$

We need to iterate through values of $u_n$ starting from $u_0$ until $u_n > M$, and return that $n$.

Pseudocode:

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def find_n_for_M(M):
    u = 1
    n = 0
    while u <= M:
        u = 1.1 * u + 3
        n += 1
    return n

4-8. Additional tasks:

4. Implement the algorithm in your calculator – This involves writing the code on a calculator or programming tool.
5. Determine from which rank $u_n > 1000$
6. Determine from which rank $u_n > 10000$
7. Determine from which rank $u_n > 10^6$
8. Determine from which rank $u_n > 10^9$

Each task involves iterating through the sequence and finding the first $n$ such that the condition $u_n > M$ is satisfied for each $M$.

Would you like me to run these calculations for specific values? Let me know if you have any questions!

Here are 5 follow-up questions:

1. What behavior do you expect from the sequence as $n$ increases?
2. Can the algorithm be optimized further for large $n$?
3. How would you modify the recurrence relation if the growth factor changed from 1.1 to another value?
4. What are the practical applications of this type of sequence?
5. How does this recursive relationship relate to geometric sequences?

Tip: Iterating large sequences can be sped up using logarithmic methods if you seek approximate values instead of exact recurrence relations.