Write the recursive formula of the following sequences use the generated patterns show your solution. 1.) 3,8,13,18,23,...  2.) 12,16,20,24,28,...

To find the recursive formulas for the given sequences, we'll identify the pattern and derive the formula based on that pattern.

### Sequence 1: 3, 8, 13, 18, 23, ...

**Step-by-step solution:**

1. **Identify the first term (\(a_1\)):**
   - The first term \(a_1 = 3\).

2. **Determine the common difference (d):**
   - The difference between consecutive terms is constant.
   - \(d = 8 - 3 = 5\)

3. **Formulate the recursive formula:**
   - The recursive formula for an arithmetic sequence is given by:
     \[
     a_{n} = a_{n-1} + d
     \]
   - Here, \(d = 5\).
   - Therefore, the recursive formula is:
     \[
     a_{n} = a_{n-1} + 5
     \]

4. **Include the initial term:**
   - The initial term is \(a_1 = 3\).

**Final recursive formula:**
\[
\begin{cases}
a_1 = 3 \\
a_{n} = a_{n-1} + 5 \quad \text{for } n > 1
\end{cases}
\]

### Sequence 2: 12, 16, 20, 24, 28, ...

**Step-by-step solution:**

1. **Identify the first term (\(a_1\)):**
   - The first term \(a_1 = 12\).

2. **Determine the common difference (d):**
   - The difference between consecutive terms is constant.
   - \(d = 16 - 12 = 4\)

3. **Formulate the recursive formula:**
   - The recursive formula for an arithmetic sequence is given by:
     \[
     a_{n} = a_{n-1} + d
     \]
   - Here, \(d = 4\).
   - Therefore, the recursive formula is:
     \[
     a_{n} = a_{n-1} + 4
     \]

4. **Include the initial term:**
   - The initial term is \(a_1 = 12\).

**Final recursive formula:**
\[
\begin{cases}
a_1 = 12 \\
a_{n} = a_{n-1} + 4 \quad \text{for } n > 1
\end{cases}
\]

Would you like more details or have any questions?

Here are 8 related questions to expand your understanding:
1. How do you derive the explicit formula for an arithmetic sequence?
2. What is the 10th term of the sequence 3, 8, 13, 18, 23, ...?
3. What is the 15th term of the sequence 12, 16, 20, 24, 28, ...?
4. How do you convert a recursive formula into an explicit formula?
5. What is the sum of the first 20 terms of the sequence 3, 8, 13, 18, 23, ...?
6. What is the sum of the first 15 terms of the sequence 12, 16, 20, 24, 28, ...?
7. How can you determine if a sequence is arithmetic?
8. What is the difference between an arithmetic and a geometric sequence?

**Tip:** When working with sequences, always verify the common difference or ratio to ensure the sequence type and the correctness of your formulas.

Discover the recursive formulas for the arithmetic sequences: 3, 8, 13, 18, 23 and 12, 16, 20, 24, 28. Step-by-step solutions provided for each sequence, including the identification of initial terms and common differences. Suitable for Grades 7-9.

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