Given the recursively defined formula a(base of expression)1=−3 , a(base of expression)n=a(base of expression)n−1 +4, determine which option is the correct function notation.(1 point) Responses

Question

f(1)=−3; f(n)=f(an−1)+4
a(1)=−3; f(n)=f(n−1)+4
f(1)=−3; f(n)=an−1+4
f(1)=−3; f(n)=f(n−1)+4

Answer 1

To understand the recursive definition given:

1. \( a(1) = -3 \)
2. \( a(n) = a(n-1) + 4 \)

This defines a sequence \( a(n) \) starting with \( a(1) = -3 \) and each subsequent term being generated by adding 4 to the previous term.

We can derive the first few terms of the sequence:

• \( a(1) = -3 \)
• \( a(2) = a(1) + 4 = -3 + 4 = 1 \)
• \( a(3) = a(2) + 4 = 1 + 4 = 5 \)
• \( a(4) = a(3) + 4 = 5 + 4 = 9 \)

So the sequence appears to be an arithmetic sequence where the first term is -3 and the common difference is 4.

Now, the function notation must reflect the recursive relationship established in the problem. The options provided can be analyzed:

1. Option 1: \( f(1) = -3; f(n) = f(an−1) + 4 \) (Incorrect, as the recursive function definition should refer to \( f(n-1) \), not \( f(a(n-1)) \)).
2. Option 2: \( a(1) = -3; f(n) = f(n−1) + 4 \) (This is incorrect, as we want a single notation for \( f \)).
3. Option 3: \( f(1) = -3; f(n) = a(n-1) + 4 \) (Incorrect, as \( f(n) \) should define a recursive relationship referring to \( f(n-1) \)).
4. Option 4: \( f(1) = -3; f(n) = f(n-1) + 4 \) (This is the correct option because it accurately states the initial condition and the recursive relationship).

Thus, the correct answer is:

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