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The question asks to express \( u_{n+1} - u_n \) in terms of \( u_n \), given the recurrence relation:

\[
u_0 = -1
\]
\[
u_{n+1} = -8 + 7u_n
\]

To find \( u_{n+1} - u_n \), we simply subtract \( u_n \) from both sides of the recurrence relation:

\[
u_{n+1} - u_n = (-8 + 7u_n) - u_n
\]

Simplifying the right-hand side:

\[
u_{n+1} - u_n = -8 + 7u_n - u_n
\]
\[
u_{n+1} - u_n = -8 + 6u_n
\]

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

\[
u_{n+1} - u_n = -8 + 6u_n
\]

Would you like further details or have any questions?

Here are 5 related questions to expand the topic:
1. How can you find an explicit formula for \( u_n \) from this recurrence relation?
2. What does the behavior of this sequence suggest as \( n \to \infty \)?
3. How would the solution change if the recurrence was \( u_{n+1} = a + bu_n \)?
4. Can you find the general term \( u_n \) by solving this as a linear recurrence?
5. How does changing the initial condition \( u_0 \) affect the sequence?

**Tip:** To solve a recurrence relation, it's helpful to identify if it's linear and if you can find a closed-form expression for general terms.

Soit la suite (u_n) avec u_0 = -1 et u_{n+1} = -8 + 7u_n. Exprimer u_{n+1} - u_n en fonction de u_n.

Learn how to express u_{n+1} - u_n in terms of u_n from a recurrence relation. This algebraic problem involves simplifying the difference between terms in the sequence based on the relation u_{n+1} = -8 + 7u_n. Suitable for Grades 10-12.

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