To solve this, we need to identify the pattern in the sequence and derive both explicit and recursive formulas.

Given Sequence:

4, 11, 18, 25, 32, ...

Step 1: Identify the Pattern

The difference between consecutive terms is constant:

• $11 - 4 = 7$
• $18 - 11 = 7$
• $25 - 18 = 7$
• $32 - 25 = 7$

The sequence is an arithmetic sequence with a common difference ($d$) of $7$.

---

Part 1: Explicit Formula

The explicit formula for an arithmetic sequence is: $a_n = a_1 + (n - 1) \cdot d$ Where:

• $a_1 = 4$ (the first term),
• $d = 7$ (the common difference),
• $n$ is the term number.

Substituting values: $a_n = 4 + (n - 1) \cdot 7$

Simplify: $a_n = 4 + 7n - 7$ $a_n = 7n - 3$

Explicit Formula: $a_n = 7n - 3$

---

Part 2: Recursive Formula

The recursive formula for an arithmetic sequence is: $a_n = a_{n-1} + d$ Where:

• $d = 7$ (the common difference),
• $a_1 = 4$ (the first term).

Recursive Formula: $a_n = a_{n-1} + 7 \quad \text{for } n \geq 2$ $a_1 = 4$

---

Final Answer:

1. Explicit Formula: $a_n = 7n - 3$

2. Recursive Formula: $a_n = a_{n-1} + 7, \quad a_1 = 4$

Let me know if you want a more detailed explanation or examples!

Related Questions:

1. What is the 10th term of this sequence?
2. How do you derive the explicit formula for a general arithmetic sequence?
3. How does the common difference affect the shape of the sequence graph?
4. What happens if the sequence starts with a different first term?
5. Can you represent this sequence graphically?

Tip:

Always verify your formulas by plugging in the first few terms of the sequence to ensure they match!