To find the explicit rule $f(n) = a + d(n - 1)$ for the sequence of row lengths, where $f(n)$ represents the row length based on the number of grocery carts $n$, we need to determine:

1. The initial row length when $n = 1$, which will be $a$.
2. The common difference $d$, which represents the rate of increase in row length for each additional cart.

Step 1: Identify the Initial Term $a$

From the graph:

• When $n = 1$ (1 cart), the row length $f(1)$ is 30. So, $a = 30$.

Step 2: Determine the Common Difference $d$

We can calculate $d$ by finding the difference between consecutive row lengths. For example:

• From $n = 1$ to $n = 2$: Row length changes from 30 to 46, so $d = 46 - 30 = 16$.
• Checking from $n = 2$ to $n = 3$: Row length changes from 46 to 62, so again $d = 62 - 46 = 16$.

Thus, the common difference $d = 16$.

Step 3: Write the Explicit Rule

Using the formula $f(n) = a + d(n - 1)$, we substitute $a = 30$ and $d = 16$: $f(n) = 30 + 16(n - 1)$ Simplifying, this becomes: $f(n) = 30 + 16n - 16 = 16n + 14$

Final Answer

The explicit rule is: $f(n) = 16n + 14$

Would you like further details on any of these steps?

Related Questions

1. What would the row length be for 10 grocery carts?
2. How would the rule change if the initial row length was different?
3. How can we use this rule to find the number of carts needed to reach a row length of 100?
4. What is the significance of the common difference in this context?
5. How does this arithmetic sequence compare to a geometric sequence?

Tip

When identifying a rule from a sequence, check for a consistent difference (for arithmetic) or ratio (for geometric) between terms to determine the type of sequence.