Let's go through the problem set you've shared. Here's a breakdown of the questions:

1. Recursive Sequence:

• The sequence is defined as $f(1) = 10$, and $f(n) = f(n-1) - 1.5$ for $n \geq 2$.

  a. Is this sequence arithmetic, geometric, or neither?

  • The sequence is arithmetic, because it has a constant difference of -1.5 between consecutive terms.

  b. List at least the first five terms of the sequence.

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

  So, the first five terms are: 10, 8.5, 7, 5.5, and 4.

  c. Graph the value of $f(n)$ as a function of term number $n$.

  • You would plot $n$ on the x-axis and the corresponding values of $f(n)$ on the y-axis, showing a downward slope because the sequence is decreasing by 1.5 each time.

2. Arithmetic Sequence $k$:

• The sequence starts at 12, 6, ...

  a. Write a recursive definition for this sequence.

  • $k(1) = 12$, and $k(n) = k(n-1) - 6$ for $n \geq 2$.

  b. Graph at least the first five terms of the sequence.

  • First five terms are: 12, 6, 0, -6, -12.
  • This would be plotted similarly to the first sequence, but with a constant difference of -6.

3. Arithmetic Sequence $a$:

• The sequence starts at 11, 7, ...

  a. Write a recursive definition for this sequence using function notation.

  • $a(1) = 11$, and $a(n) = a(n-1) - 4$ for $n \geq 2$.

  b. Sketch a graph of the first five terms of $a$.

  • First five terms are: 11, 7, 3, -1, -5.
  • You would plot these values similarly as in the previous questions.

  c. Explain how to use the recursive definition to find $a(100)$.

  • Using the recursive definition, $a(100) = a(99) - 4$, and you would continue subtracting 4 from the previous term. This method allows you to compute $a(100)$ without needing an explicit formula, but it requires knowing each preceding term.

Let me know if you want to go over any part in more detail or have questions. Here are five related questions for further exploration:

1. How do you derive an explicit formula for arithmetic sequences?
2. What is the general form of a geometric sequence, and how does it differ from arithmetic sequences?
3. How can you identify a recursive relationship by observing a sequence's terms?
4. Can you convert the recursive form of an arithmetic sequence into an explicit formula?
5. How would the graph of a geometric sequence compare to that of an arithmetic sequence?

Tip: When graphing a sequence, keep in mind that the points should not be connected since sequences are discrete functions.