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The problem asks to express \( u_{n+1} - u_n \) as a function of \( n \), where the sequence \( (u_n) \) is defined by:

\[
u_n = -8 \left( \frac{-1}{8} \right)^n
\]

### Step-by-step solution:

1. **Expression for \( u_{n+1} \):**

   The general term is given as:
   \[
   u_n = -8 \left( \frac{-1}{8} \right)^n
   \]
   Now, we need to calculate \( u_{n+1} \):
   \[
   u_{n+1} = -8 \left( \frac{-1}{8} \right)^{n+1}
   \]

2. **Simplify \( u_{n+1} \):**

   Using the property of exponents \( \left( \frac{-1}{8} \right)^{n+1} = \left( \frac{-1}{8} \right)^n \times \left( \frac{-1}{8} \right) \), we have:
   \[
   u_{n+1} = -8 \left( \frac{-1}{8} \right)^n \times \left( \frac{-1}{8} \right)
   \]
   Simplifying:
   \[
   u_{n+1} = -8 \left( \frac{-1}{8} \right)^n \times \frac{-1}{8} = \frac{8}{8} \left( \frac{-1}{8} \right)^n = \left( \frac{-1}{8} \right)^n
   \]

3. **Expression for \( u_{n+1} - u_n \):**

   Now, we subtract \( u_n \) from \( u_{n+1} \):
   \[
   u_{n+1} - u_n = \left( \frac{-1}{8} \right)^n - \left[ -8 \left( \frac{-1}{8} \right)^n \right]
   \]
   This simplifies as follows:
   \[
   u_{n+1} - u_n = \left( \frac{-1}{8} \right)^n + 8 \left( \frac{-1}{8} \right)^n
   \]
   Factor out \( \left( \frac{-1}{8} \right)^n \):
   \[
   u_{n+1} - u_n = \left( \frac{-1}{8} \right)^n \left( 1 + 8 \right)
   \]
   Therefore, the final result is:
   \[
   u_{n+1} - u_n = 9 \left( \frac{-1}{8} \right)^n
   \]

Thus, the expression for \( u_{n+1} - u_n \) in terms of \( n \) is:

\[
u_{n+1} - u_n = 9 \left( \frac{-1}{8} \right)^n
\]

Would you like further details or have any questions about this solution?

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Here are 5 related questions to expand your understanding:
1. How would the expression change if \( u_n = -7 \left( \frac{-1}{7} \right)^n \)?
2. Can this sequence \( u_n \) be classified as a geometric progression?
3. What is the common ratio of this sequence \( u_n = -8 \left( \frac{-1}{8} \right)^n \)?
4. How does the value of \( u_{n+1} - u_n \) behave as \( n \) increases?
5. What is the sum of the first \( n \) terms of the sequence \( u_n \)?

**Tip**: When working with sequences defined by powers, identifying whether it is geometric helps in simplifying the terms effectively.

Soit la suite (u_n) définie sur ℕ telle que u_n = -8 * ((-1)/8)^n. Exprimer u_{n+1} - u_n en fonction de n.

Solve for u_{n+1} - u_n in a sequence where u_n = -8 * ((-1)/8)^n. Learn the step-by-step solution to express the difference in terms of n, utilizing geometric progression concepts.

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