1. Let's start by understanding the problem: You want to know how to think about a formula involving $a_2 = 3$ without knowing the triangular formula. 2. The triangular formula usually refers to the formula for the $n$-th triangular number: $$T_n = \frac{n(n+1)}{2}$$ which counts the number of dots that can form an equilateral triangle. 3. If you don't know this formula, you can think about the sequence of triangular numbers as the sum of the first $n$ natural numbers: $$T_n = 1 + 2 + 3 + \cdots + n$$ 4. For example, $a_2 = 3$ means the second term in the sequence is 3, which matches $T_2 = 1 + 2 = 3$. 5. To find $a_n$ without the formula, you can add numbers step-by-step: - $a_1 = 1$ - $a_2 = a_1 + 2 = 3$ - $a_3 = a_2 + 3 = 6$ - and so on. 6. Alternatively, you can derive the formula by noticing the pattern: $$a_n = a_{n-1} + n$$ 7. This is a recursive definition that builds the sequence without needing the closed formula. 8. To summarize, even if you don't know the triangular formula, you can think of $a_n$ as the sum of the first $n$ natural numbers or use the recursive relation $a_n = a_{n-1} + n$ starting from $a_1 = 1$. This approach helps you understand and compute terms like $a_2 = 3$ intuitively.