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.