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The text in the image provides two sequences \( (u_n) \) and \( (v_n) \) with specific definitions and an assertion about \( (v_n) \). Here's the breakdown:

**Given:**
- \( u_0 = a \), for some real number \( a \),
- The sequence \( (u_n) \) is defined by the recurrence relation:
  \[
  u_{n+1} = \frac{1}{3} \sqrt{u_n^2 + 8}
  \]
  for all natural \( n \).

- The sequence \( (v_n) \) is defined as:
  \[
  v_n = u_n - 1 \quad \text{for all natural } n.
  \]

**Assertion:**
The sequence \( (v_n) \) is a geometric sequence.

### Objective:
We need to analyze the recurrence relation and investigate whether the sequence \( (v_n) \) forms a geometric sequence.

### Approach:
1. Start by expressing \( v_n \) in terms of \( u_n \) to see if the recurrence relation for \( u_n \) translates into a geometric sequence for \( v_n \).
2. Check if the ratio between consecutive terms of \( v_n \) is constant, which is the condition for a sequence to be geometric.

Would you like to proceed with this detailed analysis, or do you have any specific questions related to this problem?

Here are some related questions to explore:
1. What is the definition of a geometric sequence?
2. How does the recurrence relation \( u_{n+1} = \frac{1}{3} \sqrt{u_n^2 + 8} \) behave as \( n \) increases?
3. How can we express the relationship between \( u_n \) and \( v_n \) more explicitly?
4. Can a recurrence relation generate a geometric sequence under certain conditions?
5. What happens to the sequence \( u_n \) as \( n \to \infty \)?

**Tip:** When analyzing sequences, always check whether the recurrence relation stabilizes to a fixed point or pattern, as this can indicate geometric behavior.

Consider a real number a. Two sequences (u_n) and (v_n) are defined by: u_0 = a, and for all natural numbers n, u_(n+1) = (1/3) * sqrt(u_n^2 + 8), v_n = u_n - 1. Assertion 2 claims that (v_n) is a geometric sequence. Is this correct?

Explore the sequences (u_n) and (v_n) defined by a recurrence relation, and analyze whether (v_n) forms a geometric sequence. Suitable for students studying advanced sequence behavior in mathematics.

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