Let's analyze each equation to see which one defines the nth term of the function $f(n)$.

Option A: $f(n) = 7 - 5(n - 1)$ for $n \geq 1$

This equation defines the nth term as: $f(n) = 7 - 5(n - 1) = 7 - 5n + 5 = 12 - 5n$ So, the function for this option is $f(n) = 12 - 5n$.

Option B: $f(n) = 7 - 5n$ for $n \geq 1$

This equation is already simplified. It defines the nth term as $f(n) = 7 - 5n$.

Option C: $f(n) = 7 \cdot 5(789)$ for $n \geq 1$

This option seems unclear or possibly incomplete. The term $5(789)$ appears to be a constant multiplication, making the equation $f(n) = 7 \cdot 5(789)$, which would result in a constant value for all $n$.

Option D: $f(n) = 7 \cdot 57$ for $n \geq 1$

This equation is a constant function, with the nth term always equal to $7 \times 57 = 399$, regardless of $n$.

Conclusion:

The correct answer is Option B: $f(n) = 7 - 5n$ for $n \geq 1$, as it presents a linear relationship for the nth term.

Would you like more details on why Option B is correct or have any further questions?

Here are 5 related questions to deepen your understanding:

1. How do you derive the nth term of an arithmetic sequence?
2. What is the significance of $n \geq 1$ in defining a sequence?
3. Can a sequence be constant, and how do you identify such a sequence?
4. How do different forms of linear equations relate to sequences?
5. What are the differences between recursive and explicit formulas for sequences?

Tip: Always check if an equation simplifies into a recognizable pattern, such as arithmetic or geometric sequences, to understand the behavior of the nth term.