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I see the problem you've shared is from a math textbook with questions related to sequences. I'll transcribe and analyze it to help you solve the exercises.

**Transcription:**

The sequence \( (u_n) \) is defined by \( u_0 = 2 \), and for all integers \( n \geq 0 \), the recurrence relation is given by:
\[
u_{n+1} = u_n + n + 1
\]
This forms a strictly increasing sequence \( (u_n) \), and the task is to complete the following:

1. **First Exercise:** There is an excerpt from a spreadsheet. What formulas should be written in cells B3 and B4 to continue and fill the terms of the sequences \( (u_n) \) and \( (v_n) \)?

Here is the table given:

\[
\begin{array}{|c|c|c|}
n & u_n & v_n \\
1 & 4 & 7 \\
2 & 7 & 12 \\
3 & 11 & 18 \\
4 & 16 & 25 \\
5 & 22 & 33 \\
6 & 29 & 42 \\
\end{array}
\]

2. **Second Exercise:** 
- Conjecture the nature of the sequence \( (u_n) \), and find a formula for \( u_n \) as a function of \( n \) (only in terms of \( n \)).
- Conjecture and find an expression for \( v_n \), and show that it is unique.

3. **Third Exercise:** Demonstrate the formula conjectured for \( u_n \) using mathematical induction.

---

### Solution Analysis:

1. **Completion of Table for \( u_n \) and \( v_n \):**
   From the recurrence relation \( u_{n+1} = u_n + n + 1 \), we can see how \( u_n \) progresses.
   
   \[
   u_1 = u_0 + 1 + 1 = 4
   \]
   \[
   u_2 = u_1 + 2 + 1 = 7
   \]
   \[
   u_3 = u_2 + 3 + 1 = 11
   \]
   And so on.

   Now, for the sequence \( v_n \), we observe that the difference between successive terms of \( v_n \) is increasing by 1 unit more than the previous difference (a pattern commonly seen in quadratic sequences). The relation for \( v_n \) likely follows a similar quadratic expression.

2. **Conjecturing a General Formula for \( u_n \):**
   We can derive a closed formula for \( u_n \) by summing the recurrence relation:
   \[
   u_{n+1} = u_n + (n+1)
   \]
   This is the sum of the first \( n \) integers. Hence, \( u_n \) can be expressed as:
   \[
   u_n = 2 + \sum_{k=1}^{n} (k + 1) = 2 + \left(\frac{n(n+1)}{2}\right)
   \]
   Simplifying this, we get:
   \[
   u_n = 2 + \frac{n(n+1)}{2} = \frac{n(n+1)}{2} + 2
   \]
   
   This gives a quadratic expression for \( u_n \).

3. **Inductive Proof for \( u_n \):**
   To prove this formula using induction, we proceed as follows:

   **Base Case:**
   For \( n = 0 \), the formula gives:
   \[
   u_0 = 2
   \]
   which matches the initial condition.

   **Inductive Step:**
   Assume the formula holds for \( n = k \), i.e., \( u_k = \frac{k(k+1)}{2} + 2 \). Then for \( n = k+1 \):
   \[
   u_{k+1} = u_k + (k+1)
   \]
   Substituting the inductive hypothesis:
   \[
   u_{k+1} = \left(\frac{k(k+1)}{2} + 2\right) + (k+1)
   \]
   Simplifying:
   \[
   u_{k+1} = \frac{k(k+1)}{2} + (k+1) + 2 = \frac{k(k+1) + 2(k+1)}{2} + 2 = \frac{(k+1)(k+2)}{2} + 2
   \]
   Hence, the formula holds for \( n = k+1 \), completing the induction.

Would you like me to go further in any specific part of this explanation, or perhaps focus on the sequence \( v_n \) as well?

Here are **5 related questions** to help you explore further:
1. How does the recurrence relation for \( u_n \) compare to arithmetic progressions?
2. What are some common techniques for solving recurrence relations like this one?
3. How could the sum of the first \( n \) integers be derived geometrically?
4. What is the difference between linear and quadratic recurrence relations?
5. Could we generalize this sequence for any other starting value of \( u_0 \)?

**Tip:** When deriving closed formulas from recurrence relations, look for patterns in sums and try to express them using known series formulas.

Consider the sequence defined by u0 = 2, and for any integer n ≥ 0, the relation u(n+1) = u_n + n + 1. Complete the following tasks: 1. Fill the table using the recurrence relation. 2. Conjecture a formula for the sequence u_n in terms of n. 3. Prove the formula using induction.

Explore a sequence defined by the recurrence relation u(n+1) = u_n + n + 1 with step-by-step guidance. Learn to fill a sequence table, conjecture a general formula for u_n, and prove it using induction. This problem is suitable for students in Grades 9-12, covering summation and mathematical induction.

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