Question

Daniela received
\[\$30\] as a birthday present and decided to save the money to buy a new video game console. From the following week onward, she added
\[\$12\] to her savings each week.
Let
\[f(n)\] be Daniela's savings
\[n\] weeks after her birthday.
\[f\] is a sequence. What kind of sequence is it?
Choose 1 answer:
Choose 1 answer:
(Choice A) Arithmetic sequence
A
Arithmetic sequence
(Choice B) Geometric sequence
B
Geometric sequence
Complete the recursive formula for
\[f(n)\].
\[f(1)=\]
\[f(n)=f(n-1)\]

Answer 1

Daniela's savings can be modeled as an arithmetic sequence because she starts with an initial amount of money and adds a fixed amount to her savings each week.

To define the savings more explicitly:

1. She starts with \$30, so \(f(0) = 30\) (when \(n = 0\)).
2. Every week, she adds \$12 to her current savings, which represents the common difference in the arithmetic sequence.

Based on this information:

- For \(n = 1\) (after one week), she will have:
\[
f(1) = f(0) + 12 = 30 + 12 = 42
\]

So, \(f(1) = 42\).

Now, we can complete the recursive formula for \(f(n)\):
- Since \(f(n) = f(n-1) + 12\), we can fill in the recursive formula using the initial condition.

The completed recursive formula is:
- \(f(1) = 42\)
- \(f(n) = f(n-1) + 12\) for \(n \geq 1\)

Thus, the answers are:
- \(f(1) = 42\)
- \(f(n) = f(n-1) + 12\)