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Here is a step-by-step breakdown and solution to the questions in the image:

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**6. Recursive Definition for a Sequence**

The sequence is defined recursively:
- \( f(0) = -8 \)
- \( f(n) = f(n-1) + 4 \) for \( n \geq 1 \)

There is also an explicit formula for the sequence:
- \( f(n) = -8 + 4 \cdot n \) for \( n \geq 0 \)

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### Part a: Explain how these definitions represent the same sequence.
To show that the recursive and explicit formulas represent the same sequence:

1. **Start with the recursive definition**:
   - For \( n = 1 \), \( f(1) = f(0) + 4 = -8 + 4 = -4 \).
   - For \( n = 2 \), \( f(2) = f(1) + 4 = -4 + 4 = 0 \), and so on.
   - This pattern adds 4 to the previous term each time.

2. **Compare with the explicit formula**:
   - Plug \( n = 0 \) into \( f(n) = -8 + 4n \): \( f(0) = -8 + 4(0) = -8 \), matching the base case.
   - For \( n = 1 \), \( f(1) = -8 + 4(1) = -4 \), and so on. It matches the terms generated recursively.

**Conclusion**: Both definitions generate the same sequence because they produce identical terms for all \( n \geq 0 \).

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### Part b: Select a definition to calculate \( f(10) \), and explain why.
To calculate \( f(10) \):
- Using the explicit formula \( f(n) = -8 + 4n \):
  \[
  f(10) = -8 + 4(10) = -8 + 40 = 32
  \]

**Reason for choice**: The explicit formula is faster because it directly computes the term without needing to calculate all previous terms, unlike the recursive formula.

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**7. Arithmetic Sequence:**
The sequence \( m \) starts at 18, 22, ...

This is an arithmetic sequence where:
- First term (\( a \)) = 18
- Common difference (\( d \)) = 22 - 18 = 4

To find the \( 200 \)-th term:
Use the formula for the \( n \)-th term of an arithmetic sequence:
\[
m_n = a + (n-1)d
\]
Substitute \( n = 200 \), \( a = 18 \), and \( d = 4 \):
\[
m_{200} = 18 + (200-1)(4) = 18 + 199(4) = 18 + 796 = 814
\]

**Conclusion**: The 200th term is \( m_{200} = 814 \).

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Would you like a detailed explanation of any specific part? Here are some related questions for practice:

1. How can you derive the explicit formula from a recursive sequence definition?
2. What is the formula for the sum of the first \( n \) terms in an arithmetic sequence?
3. If the common difference in an arithmetic sequence changes, how does it affect the terms?
4. Can you write the explicit formula for a sequence if given only a few terms?
5. Why is the explicit formula often preferred over the recursive definition for large values of \( n \)?

**Tip**: When working with sequences, always verify base cases to ensure consistency between recursive and explicit formulas!

6. Here is a recursive definition for a sequence:

f(0) = -8,
f(n) = f(n-1) + 4 for n ≥ 1.

The explicit definition for the nth term is:
f(n) = -8 + 4n for n ≥ 0.

a. Explain how you know that these definitions represent the same sequence.
b. Select a definition to calculate f(10), and explain why you chose it.

7. An arithmetic sequence m starts 18, 22, ... Explain how you would calculate the value of the 200th term.

Learn how to compare recursive and explicit definitions of sequences and calculate specific terms. This problem covers key algebraic concepts such as recursive sequences, explicit formulas, and arithmetic sequences. Suitable for Grades 8-10.

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