To analyze this problem, let’s break it down step-by-step:

Problem Breakdown

1. Function Definition:

   • We are given a recursive function $f(n)$ defined for integers $n \geq 3$.
   • Initial condition: $f(3) = 1$.
   • Recursive formula: $f(n) = \frac{(n-1)(n-2)}{2} + f(n-1)$ for $n > 3$.

2. Objective:

   • We need to determine which of the candidate functions (not shown in the image) defined on integers $n \geq 3$ match this function $f$.

Step-by-Step Analysis

Let's calculate the values of $f(n)$ for a few terms to recognize any potential pattern.

Calculation of Terms

1. For $n = 3$: $f(3) = 1$

2. For $n = 4$: $f(4) = \frac{(4-1)(4-2)}{2} + f(3) = \frac{3 \cdot 2}{2} + 1 = 3 + 1 = 4$

3. For $n = 5$: $f(5) = \frac{(5-1)(5-2)}{2} + f(4) = \frac{4 \cdot 3}{2} + 4 = 6 + 4 = 10$

4. For $n = 6$: $f(6) = \frac{(6-1)(6-2)}{2} + f(5) = \frac{5 \cdot 4}{2} + 10 = 10 + 10 = 20$

Observing the Pattern

The function values we've computed are:

• $f(3) = 1$
• $f(4) = 4$
• $f(5) = 10$
• $f(6) = 20$

These values align with the sequence of triangular numbers: $T_n = \frac{n(n+1)}{2}$ shifted by one position. Thus, we might hypothesize that $f(n)$ follows the pattern of triangular numbers but offset by one index.

Conjecture

From our calculations, it appears that: $f(n) = T_{n-2} = \frac{(n-2)(n-1)}{2}$

To confirm, we would check that this formula satisfies the recurrence relation given in the problem. If it does, then the closed form we conjectured is likely correct.

Would you like further steps in verifying this closed-form solution or additional details on this approach?

Relative Questions:

1. How would you derive a general formula from a recursive function?
2. Can you explain why triangular numbers appear in this recursive sequence?
3. How does a recursive function differ from a closed-form function?
4. What are other techniques to solve or identify patterns in recursive sequences?
5. How do initial conditions affect the solution of a recursive function?

Tip:

When dealing with recursive functions, calculating the first few terms can often reveal useful patterns or suggest a closed form.