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The question in the image consists of a recurrence relation and two parts related to it.

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### Question 2
Consider the recurrence relation:

\[
u_n = 5u_{n-1} - 6u_{n-2}, \quad u_0 = 0, \, u_1 = 1
\]

#### (a) Find an expression for \( u_n \) in terms of \( n \). 

#### (b) For every prime number \( p > 3 \), show that \( p \mid u_{p-1} \).

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### Solution Approach:

#### Part (a): Finding an expression for \( u_n \)
This is a linear recurrence relation with constant coefficients. To solve this, we can use the characteristic equation method:

1. The characteristic equation corresponding to the recurrence relation \( u_n = 5u_{n-1} - 6u_{n-2} \) is:
   \[
   r^2 - 5r + 6 = 0
   \]
   Solve this quadratic equation to find the roots \( r_1 \) and \( r_2 \).

2. The general solution will have the form:
   \[
   u_n = A r_1^n + B r_2^n
   \]
   Use the initial conditions \( u_0 = 0 \) and \( u_1 = 1 \) to determine the constants \( A \) and \( B \).

#### Part (b): Showing \( p \mid u_{p-1} \) for every prime \( p > 3 \)
Once we have the explicit formula for \( u_n \), we need to prove that for every prime \( p > 3 \), \( p \) divides \( u_{p-1} \). This likely involves modular arithmetic and properties of primes in recurrence relations. The goal is to show that the sequence \( u_n \) satisfies a divisibility condition for prime indices.

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Would you like a detailed walkthrough of how to solve part (a), or would you like me to proceed to part (b)? Let me know!

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Here are 5 related questions:
1. What are the roots of the characteristic equation \( r^2 - 5r + 6 = 0 \)?
2. How do we apply the initial conditions to find constants \( A \) and \( B \)?
3. What is the explicit form of the solution for \( u_n \)?
4. How can recurrence relations be solved using generating functions?
5. What are some common techniques for proving divisibility in recurrence relations?

**Tip:** In recurrence relation problems, solving the characteristic equation is usually the first step to finding the general solution.

Consider the recurrence relation: u_n = 5u_(n-1) - 6u_(n-2), with u_0 = 0 and u_1 = 1. (a) Find an expression for u_n in terms of n. (b) For every prime number p > 3, show that p divides u_(p-1).

Explore the solution of a linear recurrence relation u_n = 5u_(n-1) - 6u_(n-2) with initial conditions u_0 = 0 and u_1 = 1. Solve for an explicit formula and prove prime divisibility for u_(p-1) where p > 3. Ideal for undergraduate mathematics students.

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