8,13,18,23 Write a recursive formula for each sequence.

To write a recursive formula for each of the given sequences, we need to identify the pattern or rule that defines the sequence in terms of its previous terms.

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

Let's denote the \( n \)-th term of the sequence by \( a_n \).

**First term:** \( a_1 = 8 \)

**General pattern:** Each term increases by 5 from the previous term.
\[ a_{n+1} = a_n + 5 \]

**Recursive formula:** 
\[ \begin{cases}
a_1 = 8 \\
a_{n+1} = a_n + 5 \quad \text{for } n \geq 1
\end{cases} \]

### Sequence 2: 13, 18, 23, ...

Let's denote the \( n \)-th term of the sequence by \( b_n \).

**First term:** \( b_1 = 13 \)

**General pattern:** Each term increases by 5 from the previous term.
\[ b_{n+1} = b_n + 5 \]

**Recursive formula:** 
\[ \begin{cases}
b_1 = 13 \\
b_{n+1} = b_n + 5 \quad \text{for } n \geq 1
\end{cases} \]

### Sequence 3: 18, 23, ...

Let's denote the \( n \)-th term of the sequence by \( c_n \).

**First term:** \( c_1 = 18 \)

**General pattern:** Each term increases by 5 from the previous term.
\[ c_{n+1} = c_n + 5 \]

**Recursive formula:** 
\[ \begin{cases}
c_1 = 18 \\
c_{n+1} = c_n + 5 \quad \text{for } n \geq 1
\end{cases} \]

### Sequence 4: 23, ...

Let's denote the \( n \)-th term of the sequence by \( d_n \).

**First term:** \( d_1 = 23 \)

Since there's only one term provided, we can't determine the pattern of change. Therefore, we can assume it's a constant sequence with only one term.

**Recursive formula:** 
\[ d_n = 23 \quad \text{for all } n \geq 1 \]

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Do you want any details or have any questions? Here are some related questions:

1. What is a recursive formula?
2. How do you derive a recursive formula from a sequence?
3. Can a sequence have more than one recursive formula?
4. What is the difference between a recursive formula and an explicit formula?
5. How would you find the explicit formula for the given sequences?
6. How does the initial term affect the recursive formula?
7. Can recursive formulas be used for non-arithmetic sequences?
8. How do you identify the pattern in a sequence?

**Tip:** When writing a recursive formula, always identify the initial term(s) and the rule that relates each term to the previous term(s).

Learn how to write recursive formulas for sequences: 8, 13, 18, 23. Understand the pattern of increase and derive recursive formulas step-by-step. Suitable for high school students studying sequences and recursive functions.

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