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Travailler moi cet exercice par le programme éducatif d un lyceean

Let's break down and solve the exercise in parts. Here's the step-by-step solution to each of the questions in the problem:

---

### **Given Information:**
We are given a sequence \( (u_n) \) defined on \( \mathbb{N}^* \), and the sum of the first \( n \) terms is given by:

\[
S_n = u_1 + u_2 + \dots + u_n
\]
with \( S_n = 3n^2 + 5n \).

### **1) Calculate \( u_1 \), \( u_2 \), and \( u_3 \):**

To calculate the individual terms \( u_1 \), \( u_2 \), and \( u_3 \), we need to recognize that:

\[
u_n = S_n - S_{n-1}
\]

#### Calculation of \( u_1 \):
Since \( S_1 = 3(1)^2 + 5(1) = 3 + 5 = 8 \), we have \( u_1 = S_1 = 8 \).

#### Calculation of \( u_2 \):
We need to compute \( S_2 \) and \( S_1 \) first:

\[
S_2 = 3(2)^2 + 5(2) = 12 + 10 = 22
\]
Thus:
\[
u_2 = S_2 - S_1 = 22 - 8 = 14
\]

#### Calculation of \( u_3 \):
Similarly, we calculate \( S_3 \):

\[
S_3 = 3(3)^2 + 5(3) = 27 + 15 = 42
\]
Thus:
\[
u_3 = S_3 - S_2 = 42 - 22 = 20
\]

### **Summary:**
- \( u_1 = 8 \)
- \( u_2 = 14 \)
- \( u_3 = 20 \)

---

### **2a) Express \( S_{n-1} \) as a function of \( n \):**

Since \( S_n = 3n^2 + 5n \), we can substitute \( n-1 \) in place of \( n \) to find \( S_{n-1} \):

\[
S_{n-1} = 3(n-1)^2 + 5(n-1)
\]
Expanding this expression:

\[
S_{n-1} = 3(n^2 - 2n + 1) + 5(n-1)
\]
\[
S_{n-1} = 3n^2 - 6n + 3 + 5n - 5
\]
\[
S_{n-1} = 3n^2 - n - 2
\]

---

### **2b) Express \( u_n \) as a function of \( n \):**

Now, using the definition of \( u_n = S_n - S_{n-1} \):

\[
u_n = (3n^2 + 5n) - (3n^2 - n - 2)
\]
Simplifying the expression:

\[
u_n = 3n^2 + 5n - 3n^2 + n + 2
\]
\[
u_n = 6n + 2
\]

Thus, the general term of the sequence is:

\[
u_n = 6n + 2
\]

---

### **3) Show that \( (u_n) \) is an arithmetic sequence:**

We have found that \( u_n = 6n + 2 \). This is clearly an arithmetic sequence, as it is in the form \( u_n = u_1 + (n-1) \times r \), where:
- The first term \( u_1 = 8 \),
- The common difference (reason) \( r = 6 \) (since \( u_{n+1} - u_n = 6 \)).

Thus, the sequence \( (u_n) \) is an arithmetic sequence with:
- **First term** \( u_1 = 8 \),
- **Common difference** \( r = 6 \).

---

### **4) Graphical representation:**

The final part asks for a graphical representation of the sequences \( (u_n) \) and \( (S_n) \) on the same coordinate system.

- The sequence \( u_n = 6n + 2 \) is linear, with slope 6.
- The sequence \( S_n = 3n^2 + 5n \) is quadratic, with a parabolic shape opening upwards.

We can plot both of these sequences for values of \( n \in \mathbb{N}^* \), but the detailed plot would typically be done using software or graphing tools.

---

### Questions to Expand Understanding:

1. What is the general formula for the sum of an arithmetic sequence, and how would you apply it to \( u_n \)?
2. How does the sequence \( S_n \) grow compared to \( u_n \) as \( n \) becomes large?
3. Can you find the value of \( u_4 \) using both the recursive and the explicit formula?
4. What is the significance of the common difference \( r \) in an arithmetic sequence?
5. How would the results change if \( S_n \) were a cubic function instead of quadratic?

---

**Tip:** In problems involving sequences and sums, finding the difference between consecutive terms is a powerful way to identify patterns, especially for arithmetic or geometric sequences.

Soit (u_n) une suite définie sur IN*. On pose S_n = u_1 + u_2 + ... + u_n et on suppose que S_n = 3n^2 + 5n. 1) Calculer u_1, u_2 et u_3. 2) a. Exprimer S_{n-1} en fonction de n b. Exprimer u_n en fonction de n 3) Montrer que (u_n) est une suite arithmétique. 4) Représenter (u_n) et (S_n).

Learn how to calculate terms in an arithmetic sequence, express functions of n, and graph sequences. This problem focuses on the sequence (u_n) and the sum S_n, with a detailed solution involving the expressions S_n = 3n^2 + 5n. Suitable for Grades 10-12.

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