To complete the recursive formula for the function $f(n) = 93 + 4(n - 1)$, we need to express it in a recursive form. A recursive formula expresses each term in a sequence as a function of the previous term(s).

The given function is: $f(n) = 93 + 4(n - 1)$ This is an explicit formula, but we can convert it into a recursive form.

Step-by-step approach:

1. Calculate the first term (Base case):

   • For $f(1)$, plug $n = 1$ into the formula: $f(1) = 93 + 4(1 - 1) = 93 + 0 = 93$
   • So, the base case is $f(1) = 93$.

2. Find a recurrence relation:

   • To express $f(n)$ in terms of $f(n - 1)$, we need to subtract $f(n - 1)$ from $f(n)$: $f(n) - f(n - 1) = \left( 93 + 4(n - 1) \right) - \left( 93 + 4(n - 2) \right)$ Simplifying the equation: $f(n) - f(n - 1) = 4(n - 1) - 4(n - 2)$ $f(n) - f(n - 1) = 4n - 4 - 4n + 8 = 4$ Therefore, the recurrence relation is: $f(n) = f(n - 1) + 4$
   • So, the recursive formula for $f(n)$ is: $f(1) = 93 \quad \text{and} \quad f(n) = f(n - 1) + 4 \quad \text{for} \quad n > 1.$

Summary:

The recursive formula for $f(n)$ is: $f(1) = 93$ $f(n) = f(n - 1) + 4 \quad \text{for} \quad n > 1.$

Would you like any further explanation or assistance with another example?

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Follow-up questions:

1. How would the recursive formula change if the original equation was $f(n) = 93 + 4n$?
2. Can you identify the common difference in this sequence?
3. What is the value of $f(5)$ using the recursive formula?
4. How would you prove that the recursive formula is equivalent to the explicit formula $f(n) = 93 + 4(n - 1)$?
5. What would the recursive formula be for a sequence defined by $f(n) = 5n + 2$?

Tip:

In general, to convert an explicit formula into a recursive formula, calculate the difference between successive terms and express it as a recurrence relation.