Find an explicit function rule and a recursive rule for the number of dots in step n. Step 1

To find an explicit function rule and a recursive rule for the number of dots in step \( n \), we first need to understand the pattern of dots in the steps. Let's assume that the pattern of dots increases in a specific way. For example, if we have a triangular arrangement of dots, the number of dots in each step can be represented as follows: - Step 1: 1 dot - Step 2: 3 dots (1 + 2) - Step 3: 6 dots (1 + 2 + 3) - Step 4: 10 dots (1 + 2 + 3 + 4) - Step 5: 15 dots (1 + 2 + 3 + 4 + 5) From this pattern, we can see that the number of dots in step \( n \) corresponds to the \( n \)-th triangular number, which can be calculated using the formula: ### Explicit Function Rule The explicit function rule for the number of dots in step \( n \) is given by: \[ d(n) = \frac{n(n + 1)}{2} \] where \( d(n) \) is the number of dots at step \( n \). ### Recursive Rule The recursive rule can be defined as follows: \[ d(1) = 1 \] \[ d(n) = d(n - 1) + n \quad \text{for } n > 1 \] This means that the number of dots in step \( n \) is equal to the number of dots in the previous step plus \( n \). ### Summary - **Explicit Function Rule**: \( d(n) = \frac{n(n + 1)}{2} \) - **Recursive Rule**: - \( d(1) = 1 \) - \( d(n) = d(n - 1) + n \) for \( n > 1 \)