Question

Given the recursively defined formula a1=−3 , an=an−1+4 , determine which option is the correct function notation.(1 point) Responses a(1)=−3 ; f(n)=f(n−1)+4 a left parenthesis 1 right parenthesis equals negative 3 ; f left parenthesis n right parenthesis equals f left parenthesis n minus 1 right parenthesis plus 4 f(1)=−3 ; f(n)=f(an−1)+4 f left parenthesis 1 right parenthesis equals negative 3 ; f left parenthesis n right parenthesis equals f left parenthesis a subscript n minus 1 baseline right parenthesis plus 4 f(1)=−3 ; f(n)=an−1+4 f left parenthesis 1 right parenthesis equals negative 3 ; f left parenthesis n right parenthesis equals a subscript n minus 1 baseline plus 4 f(1)=−3 ; f(n)=f(n−1)+4

Answer 1

To solve for the correct function notation based on the given recursively defined formula, let's first clarify the definitions provided:

1. The initial term \(a_1 = -3\).
2. The recursive relation is defined by \(a_n = a_{n-1} + 4\).

From the recursive definition, we can express this in function notation:

• We can define a function \(f(n)\) such that:
  • \(f(1) = -3\) (the initial value matches \(a_1\))
  • The recursive relationship \(f(n) = f(n-1) + 4\) (mirroring \(a_n\) definition).

Now, let's evaluate the options provided:

1. a(1) = -3 ; f(n) = f(n - 1) + 4

   • This means \(a_1 = -3\) and \(f(n) = f(n - 1) + 4\). This is not consistent as \(a\) and \(f\) should represent the same sequence.

2. f(1) = -3 ; f(n) = f(n-1) + 4

   • This correctly shows \(f(1) = -3\) and that \(f(n)\) follows the same recursive definition.

3. f(1) = -3 ; f(n) = f(a_{n-1}) + 4

   • This introduces the incorrect reference \(f(a_{n-1})\), deviating from the original recursive structure.

4. f(1) = -3 ; f(n) = a_{n - 1} + 4

   • This incorrectly changes the function's recursive nature, as it suggests direct calculation of \(a_{n-1}\).

5. f(1) = -3 ; f(n) = f(n - 1) + 4

   • Similar to option 2, but with a slight formatting difference.

Given the analysis, the correct function notation is:

f(1) = -3 ; f(n) = f(n - 1) + 4.

Thus, response option 2 is the correct option.