Let’s analyze and solve the problem.

Problem:

We are given a sequence $\{u_n\}$ defined by: $u_1 = 5, \quad u_{n+1} = u_n + n, \; \forall n \geq 1.$ We are tasked to determine which formula for $u_n$ is correct among the given options.

Step-by-step Solution:

1. Write the recurrence explicitly for a few terms: $u_1 = 5,$ $u_2 = u_1 + 1 = 5 + 1 = 6,$ $u_3 = u_2 + 2 = 6 + 2 = 8,$ $u_4 = u_3 + 3 = 8 + 3 = 11,$ $u_5 = u_4 + 4 = 11 + 4 = 15.$

   We observe that the sequence grows based on the sum of integers up to $n-1$.

2. General formula for the sum of integers: The sum of the first $k$ integers is given by: $S_k = \frac{k(k+1)}{2}.$

   For the given recurrence, the sequence $u_n$ can be expressed as: $u_n = 5 + S_{n-1}.$ Substituting $S_{n-1} = \frac{(n-1)n}{2}$, we get: $u_n = 5 + \frac{(n-1)n}{2}.$

3. Check against the options:

   • Option A: $u_n = 5 + \frac{n(n-1)}{2}$.
     This matches our derived formula.

   • Option B: $u_n = \frac{n(n-1)}{2}$.
     This is incorrect because it omits the initial value $u_1 = 5$.

   • Option C: $u_n = 5 + \frac{n(n+1)}{2}$.
     This is incorrect because it uses $n(n+1)$ instead of $n(n-1)$.

   • Option D: $u_n = 5 + \frac{(n-1)(n+2)}{2}$.
     This is incorrect because it uses $(n-1)(n+2)$ instead of $(n-1)n$.

Final Answer:

The correct formula is A: $u_n = 5 + \frac{n(n-1)}{2}$.

Would you like a detailed explanation of any part of this solution?
Here are 5 questions you might want to explore further:

1. How is the sum of consecutive integers derived?
2. Why does the recurrence formula relate to sums of integers?
3. Could you verify the solution for higher values of $n$?
4. How does the initial value $u_1 = 5$ affect the formula?
5. Can similar recurrence sequences be solved systematically?

Tip: Always expand the first few terms of a recurrence to identify patterns before attempting a general formula!