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

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 $n \geq 0$

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 $n \geq 0$.

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.

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 $m_{200} = 814$.

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!