Let's break this down:

The explicit equation given is:
$f(n) = 4n + 6$

From this, we can calculate the values in the sequence based on the term number $n$. For example:

• For $n = 1$, $f(1) = 4(1) + 6 = 10$
• For $n = 2$, $f(2) = 4(2) + 6 = 14$

The table indicates the sequence continues similarly for higher terms, like term #50.

Recursive Equation:

You need to write the recursive equation for this sequence.

The general recursive form is: $f(n) = f(n-1) + d$ where $d$ is the common difference.

Looking at the sequence values (10, 14, ...), the common difference $d$ between terms is 4: $d = 14 - 10 = 4$

Thus, the recursive formula is: $f(n) = f(n-1) + 4$

The start value (base case) is the first term: $f(1) = 10$

So, the recursive equation becomes: $f(n) = f(n-1) + 4 \text{ with } f(1) = 10$

Do you want more details or have any questions? Here are five related questions for deeper understanding:

1. How would the recursive equation change if the common difference were negative?
2. Can you derive the 50th term using both the explicit and recursive formulas?
3. How does the initial condition $f(1)$ influence the recursive sequence?
4. What is the relationship between the explicit and recursive formulas in arithmetic sequences?
5. How would you explain the difference between an arithmetic and a geometric sequence?

Tip: In arithmetic sequences, the common difference $d$ remains constant between terms, which simplifies the recursive formula significantly!