90,30,10,10/3 Write a recursive formula for each sequence.

, 8, 13, 18, 23…

, 2, −8, 32, −128…

=*a*18

=*a*12

=*an*5for≥*n*2

=*an*4for≥*n*2


### Sequence 1: 90, 30, 10, 10/3, ...

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

**First term:** \( a_1 = 90 \)

**General pattern:** Each term is one-third of the previous term.
\[ a_{n+1} = \frac{a_n}{3} \]

**Recursive formula:** 
\[ \begin{cases}
a_1 = 90 \\
a_{n+1} = \frac{a_n}{3} \quad \text{for } n \geq 1
\end{cases} \]

### Sequence 2: 8, 13, 18, 23, ...

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

**First term:** \( b_1 = 8 \)

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

**Recursive formula:** 
\[ \begin{cases}
b_1 = 8 \\
b_{n+1} = b_n + 5 \quad \text{for } n \geq 1
\end{cases} \]

### Sequence 3: 2, −8, 32, −128, ...

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

**First term:** \( c_1 = 2 \)

**General pattern:** Each term is multiplied by -4 from the previous term.
\[ c_{n+1} = -4 \cdot c_n \]

**Recursive formula:** 
\[ \begin{cases}
c_1 = 2 \\
c_{n+1} = -4 \cdot c_n \quad \text{for } n \geq 1
\end{cases} \]

### Sequence 4: \(*a*18\)

Since the sequence is given as "*a*18," let's assume this is a constant sequence where each term is 18.

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

**First term:** \( d_1 = 18 \)

**General pattern:** Each term is 18.
\[ d_{n+1} = d_n \]

**Recursive formula:** 
\[ \begin{cases}
d_1 = 18 \\
d_{n+1} = d_n \quad \text{for } n \geq 1
\end{cases} \]

### Sequence 5: \(*a*12\)

Similarly, let's assume this is a constant sequence where each term is 12.

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

**First term:** \( e_1 = 12 \)

**General pattern:** Each term is 12.
\[ e_{n+1} = e_n \]

**Recursive formula:** 
\[ \begin{cases}
e_1 = 12 \\
e_{n+1} = e_n \quad \text{for } n \geq 1
\end{cases} \]

### Sequence 6: \(*an*5\) for \( n \geq 2 \)

This appears to be an arithmetic sequence with a common difference of 5, starting from an unspecified initial term \( a \).

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

**First term:** \( f_1 = a \)

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

**Recursive formula:** 
\[ \begin{cases}
f_1 = a \\
f_{n+1} = f_n + 5 \quad \text{for } n \geq 1
\end{cases} \]

### Sequence 7: \(*an*4\) for \( n \geq 2 \)

This appears to be a geometric sequence with a common ratio of 4, starting from an unspecified initial term \( a \).

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

**First term:** \( g_1 = a \)

**General pattern:** Each term is multiplied by 4 from the previous term.
\[ g_{n+1} = 4 \cdot g_n \]

**Recursive formula:** 
\[ \begin{cases}
g_1 = a \\
g_{n+1} = 4 \cdot g_n \quad \text{for } n \geq 1
\end{cases} \]

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

1. How do you determine if a sequence is arithmetic or geometric?
2. What is the difference between a common difference and a common ratio?
3. Can a sequence be both arithmetic and geometric?
4. How do you find the explicit formula for a geometric sequence?
5. What is the significance of the first term in a recursive sequence?
6. How can you convert a recursive formula to an explicit formula?
7. What are some real-life applications of recursive sequences?
8. How do you identify patterns in non-arithmetic and non-geometric sequences?

**Tip:** Always check if the terms of a sequence follow an arithmetic or geometric pattern before writing the recursive formula. This will help you identify the correct rule for the sequence.

Learn how to write recursive formulas for arithmetic and geometric sequences with step-by-step explanations. This page covers sequences where terms follow specific patterns of addition or multiplication. Suitable for high school students.

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