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Exercise 1: Sequence $u_n$

The sequence $u_n$ is defined with $u_1 = 0.9$, $u_2 = 0.99$, $u_3 = 0.999$, and so on, where each $u_n$ consists of $n$ digits of 9 after the decimal.

1. Showing that $u_n$ is the sum of $n$ terms of a geometric sequence:

   • Each term $u_n$ can be written as: $u_n = 1 - 10^{-n}$ This is the sum of a geometric series with the first term $a = 0.9$ and common ratio $r = 0.1$.

2. Proof that $u_{n+1} = \frac{1}{10}(u_n + 9)$:

   • From the definition of $u_n = 1 - 10^{-n}$, you can derive the relation $u_{n+1} = \frac{1}{10}(u_n + 9)$.

3. Expressing $u_n$ as a function of $n$:

   • We already derived the general form: $u_n = 1 - 10^{-n}$

4. Finding the limit of $a_n$ as $n \to \infty$:

   • As $n \to \infty$, $u_n \to 1$ because the term $10^{-n}$ approaches 0.

Exercise 2: Recurrence sequence $u_n$ defined by $u_0 = 5$ and $u_{n+1} = 2u_n - 3$

1. Calculating $u_1$, $u_2$, and $u_3$:

   • Using the recurrence relation: $u_1 = 2 \cdot 5 - 3 = 7$ $u_2 = 2 \cdot 7 - 3 = 11$ $u_3 = 2 \cdot 11 - 3 = 19$

2. Plotting and drawing the sequences:

   • The points $(0, 5), (1, 7), (2, 11), (3, 19)$ can be plotted.

3. Representing the first five terms on the graph:

   • This can be done on a coordinate system, plotting the terms as discrete points.

Exercise 3: Sequence $U_n$ defined by recurrence relation

$U_0 = 3$, $U_{n+1} = -\frac{1}{2}U_n + 3$

1. Calculating $U_1$ and $U_2$:
   • Using the recurrence relation: $U_1 = -\frac{1}{2} \cdot 3 + 3 = 1.5$ $U_2 = -\frac{1}{2} \cdot 1.5 + 3 = 2.25$

Exercise 4: Sequence $u_n$ with the given recurrence relation

1. Calculating $u_1$, $u_2$ using the given recurrence relation and plotting the graph.

Would you like me to go into more detail for any specific part or to proceed with further explanations and solutions?

Here are five follow-up questions that expand on the material:

1. How can you generalize the limit for a sequence like $u_n$?
2. What methods are there to determine if a recurrence relation is geometric or arithmetic?
3. How do you prove that a sequence is neither arithmetic nor geometric?
4. How can the geometric series formula be applied to find the sum of a sequence like $u_n$?
5. What is the significance of the intersection point in Exercise 4?

Tip: Always break down a recurrence relation to check for patterns or properties like linearity or geometric behavior.