The problem you're working on presents a recursive sequence defined as follows:

$f(1) = 10, \quad f(n) = f(n-1) - 1.5 \text{ for } n \geq 2.$

Let's break this down:

Part (a)

Is this sequence arithmetic, geometric, or neither?

An arithmetic sequence has a constant difference between consecutive terms. The recursive formula $f(n) = f(n-1) - 1.5$ subtracts 1.5 from the previous term each time, which indicates a constant difference of $-1.5$. Therefore, this is an arithmetic sequence.

Part (b)

List at least the first five terms of the sequence.

Using the recursive relation:

• $f(1) = 10$
• $f(2) = f(1) - 1.5 = 10 - 1.5 = 8.5$
• $f(3) = f(2) - 1.5 = 8.5 - 1.5 = 7$
• $f(4) = f(3) - 1.5 = 7 - 1.5 = 5.5$
• $f(5) = f(4) - 1.5 = 5.5 - 1.5 = 4$

So, the first five terms of the sequence are: $10, 8.5, 7, 5.5, 4.$

Let me know if you need further details or explanations!

Here are 5 related questions:

1. How can we generalize the formula for any arithmetic sequence?
2. What would the 10th term of this sequence be?
3. How can we find the sum of the first $n$ terms of an arithmetic sequence?
4. Can a recursive formula be used to represent geometric sequences as well?
5. What are the applications of arithmetic sequences in real-world problems?

Tip: For arithmetic sequences, you can always use the formula $a_n = a_1 + (n-1)d$, where $d$ is the common difference.