Let’s go through the problems one by one. I will solve each for clarity.

1. Recursive formula to explicit formula

The given recursive formula: $b(1) = 8, \quad b(n) = b(n-1) - 6$ is an arithmetic sequence where the first term is $b(1) = 8$ and the common difference is $-6$.

Explicit formula for an arithmetic sequence: $b(n) = b(1) + (n-1)d$ Substitute $b(1) = 8$ and $d = -6$: $b(n) = 8 + (n-1)(-6)$ Simplify: $b(n) = 8 - 6(n-1)$ $b(n) = 8 - 6n + 6$ $b(n) = 14 - 6n$ Final explicit formula: $b(n) = 14 - 6n$

2. Recursive formula to explicit formula

The given recursive formula: $b(1) = 4, \quad b(n) = b(n-1) + 3$ is an arithmetic sequence where $b(1) = 4$ and $d = 3$.

Explicit formula: $b(n) = b(1) + (n-1)d$ Substitute $b(1) = 4$ and $d = 3$: $b(n) = 4 + (n-1)(3)$ Simplify: $b(n) = 4 + 3n - 3$ $b(n) = 3n + 1$ Final explicit formula: $b(n) = 3n + 1$

3. Recursive formula to explicit formula

The given recursive formula: $f(1) = 11, \quad f(n) = f(n-1) - 1$ is an arithmetic sequence where $f(1) = 11$ and $d = -1$.

Explicit formula: $f(n) = f(1) + (n-1)d$ Substitute $f(1) = 11$ and $d = -1$: $f(n) = 11 + (n-1)(-1)$ Simplify: $f(n) = 11 - n + 1$ $f(n) = 12 - n$ Final explicit formula: $f(n) = 12 - n$

4. Recursive formula to explicit formula

The given recursive formula: $g(1) = 7, \quad g(n) = g(n-1) + 5$ is an arithmetic sequence where $g(1) = 7$ and $d = 5$.

Explicit formula: $g(n) = g(1) + (n-1)d$ Substitute $g(1) = 7$ and $d = 5$: $g(n) = 7 + (n-1)(5)$ Simplify: $g(n) = 7 + 5n - 5$ $g(n) = 5n + 2$ Final explicit formula: $g(n) = 5n + 2$

5. Explicit formula to recursive formula

The given explicit formula: $a(n) = -12 + 6(n-1)$ Simplify the explicit formula first: $a(n) = -12 + 6n - 6$ $a(n) = 6n - 18$

From this explicit formula:

• The first term is: $a(1) = 6(1) - 18 = -12$
• The common difference is $6$ (the coefficient of $n$).

Recursive formula: $a(1) = -12, \quad a(n) = a(n-1) + 6$

Summary of Results

1. $b(n) = 14 - 6n$
2. $b(n) = 3n + 1$
3. $f(n) = 12 - n$
4. $g(n) = 5n + 2$
5. $a(1) = -12, \quad a(n) = a(n-1) + 6$

Let me know if you want further clarification or detailed steps for any problem!

Follow-up Questions

1. Can you derive the general explicit formula for any arithmetic sequence?
2. How do you find the common difference when switching from explicit to recursive formulas?
3. What happens if the sequence has a geometric progression instead?
4. How do you confirm your answer for recursive formulas?
5. How do arithmetic sequences relate to linear equations?

Tip: Always simplify the explicit formula first before converting it into recursive form!