Practice worksheet for writing recursive formulas of arithmetic sequences.

To solve the problems involving arithmetic sequences, we will use the recursive formula for an arithmetic sequence:

Recursive Formula for Arithmetic Sequences:

The recursive formula for an arithmetic sequence is given by:
$$
a_n = a_{n-1} + d
$$
where:
- \( a_n \) is the \( n \)-th term,
- \( a_{n-1} \) is the previous term,
- \( d \) is the common difference.

Additionally, the first term \( a_1 \) is explicitly provided.

Let's solve each problem step by step.

---

Problem 1:

Given: \( a_1 = 7 \), \( d = 3 \)

#### Step 1: Write the recursive formula.
Using the recursive formula:
$$
a_n = a_{n-1} + d
$$
Substitute \( d = 3 \):
$$
a_n = a_{n-1} + 3
$$

#### Step 2: Provide the initial condition.
The first term is given as:
$$
a_1 = 7
$$

#### Final Answer:
$$
\boxed{a_n = a_{n-1} + 3, \quad a_1 = 7}
$$

---

Problem 2:

Given: \( a_1 = -60 \), \( d = 5 \)

#### Step 1: Write the recursive formula.
Using the recursive formula:
$$
a_n = a_{n-1} + d
$$
Substitute \( d = 5 \):
$$
a_n = a_{n-1} + 5
$$

#### Step 2: Provide the initial condition.
The first term is given as:
$$
a_1 = -60
$$

#### Final Answer:
$$
\boxed{a_n = a_{n-1} + 5, \quad a_1 = -60}
$$

---

Problem 3:

Given: \( a_1 = 9.5 \), \( d = 0.25 \)

#### Step 1: Write the recursive formula.
Using the recursive formula:
$$
a_n = a_{n-1} + d
$$
Substitute \( d = 0.25 \):
$$
a_n = a_{n-1} + 0.25
$$

#### Step 2: Provide the initial condition.
The first term is given as:
$$
a_1 = 9.5
$$

#### Final Answer:
$$
\boxed{a_n = a_{n-1} + 0.25, \quad a_1 = 9.5}
$$

---

Problem 4:

Given: \( a_1 = 100 \), \( d = -10 \)

#### Step 1: Write the recursive formula.
Using the recursive formula:
$$
a_n = a_{n-1} + d
$$
Substitute \( d = -10 \):
$$
a_n = a_{n-1} - 10
$$

#### Step 2: Provide the initial condition.
The first term is given as:
$$
a_1 = 100
$$

#### Final Answer:
$$
\boxed{a_n = a_{n-1} - 10, \quad a_1 = 100}
$$

---

Problem 5:

Given: \( a_1 = 2 \), \( d = 2 \)

#### Step 1: Write the recursive formula.
Using the recursive formula:
$$
a_n = a_{n-1} + d
$$
Substitute \( d = 2 \):
$$
a_n = a_{n-1} + 2
$$

#### Step 2: Provide the initial condition.
The first term is given as:
$$
a_1 = 2
$$

#### Final Answer:
$$
\boxed{a_n = a_{n-1} + 2, \quad a_1 = 2}
$$

---

Problem 6:

Given: \( a_1 = 81 \), \( d = -\frac{1}{3} \)

#### Step 1: Write the recursive formula.
Using the recursive formula:
$$
a_n = a_{n-1} + d
$$
Substitute \( d = -\frac{1}{3} \):
$$
a_n = a_{n-1} - \frac{1}{3}
$$

#### Step 2: Provide the initial condition.
The first term is given as:
$$
a_1 = 81
$$

#### Final Answer:
$$
\boxed{a_n = a_{n-1} - \frac{1}{3}, \quad a_1 = 81}
$$

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Final Summary of Answers:

1. \( \boxed{a_n = a_{n-1} + 3, \quad a_1 = 7} \)
2. \( \boxed{a_n = a_{n-1} + 5, \quad a_1 = -60} \)
3. \( \boxed{a_n = a_{n-1} + 0.25, \quad a_1 = 9.5} \)
4. \( \boxed{a_n = a_{n-1} - 10, \quad a_1 = 100} \)
5. \( \boxed{a_n = a_{n-1} + 2, \quad a_1 = 2} \)
6. \( \boxed{a_n = a_{n-1} - \frac{1}{3}, \quad a_1 = 81} \)

Each answer provides the recursive formula and the initial condition for the respective arithmetic sequence.