Let's solve the problems from the image step by step:

Problem 2

Recursive formula: $b(1) = -4, \quad b(n) = b(n-1) + 9$

Solution: The recursive formula represents an arithmetic sequence with:

• First term: $b(1) = -4$
• Common difference: $d = 9$

The explicit formula for an arithmetic sequence is: $b(n) = b(1) + (n-1)d$

Substituting $b(1) = -4$ and $d = 9$: $b(n) = -4 + (n-1)(9)$ Simplify: $b(n) = -4 + 9n - 9$ $b(n) = 9n - 13$

Explicit formula: $b(n) = 9n - 13$

Problem 4

Recursive formula: $f(1) = -7, \quad f(n) = f(n-1) + 2$

Solution: The recursive formula represents an arithmetic sequence with:

• First term: $f(1) = -7$
• Common difference: $d = 2$

The explicit formula for an arithmetic sequence is: $f(n) = f(1) + (n-1)d$

Substituting $f(1) = -7$ and $d = 2$: $f(n) = -7 + (n-1)(2)$ Simplify: $f(n) = -7 + 2n - 2$ $f(n) = 2n - 9$

Explicit formula: $f(n) = 2n - 9$

Problem 6

Explicit formula: $g(n) = 8n - 6$

Convert this to a recursive formula:

1. The common difference ($d$) can be determined by observing the explicit formula. The term-to-term difference in an arithmetic sequence is the coefficient of $n$, which is $d = 8$.

2. The recursive formula starts with the first term: $g(1) = 8(1) - 6 = 2$

3. The recursive formula is: $g(n) = g(n-1) + 8 \quad \text{for } n \geq 2$

Recursive formula: $g(1) = 2, \quad g(n) = g(n-1) + 8$

Summary of Results

1. Problem 2: $b(n) = 9n - 13$
2. Problem 4: $f(n) = 2n - 9$
3. Problem 6: $g(1) = 2, \quad g(n) = g(n-1) + 8$

Do you have any questions or need further clarification on any of these problems?

Expanding Questions

1. How do we derive the general formula for an arithmetic sequence?
2. Can we apply this technique to geometric sequences?
3. What is the significance of the first term and common difference in these sequences?
4. How can recursive and explicit formulas be used interchangeably in real-world problems?
5. How would the formulas change if the sequences were geometric instead of arithmetic?

Tip: Always verify your sequence terms by plugging in values from your explicit or recursive formula to ensure consistency!