Sure! Below, I will provide three different recursive equations with their step-by-step solutions.

1. Arithmetic Sequence:

Recursive equation: $a_1 = 5 \quad \text{and} \quad a_n = a_{n-1} + 3 \text{ for } n > 1$

Step-by-step solution:

• Given: $a_1 = 5$ and the recursive formula $a_n = a_{n-1} + 3$.
• To find the first few terms of the sequence, we start from $a_1$.

Finding the terms: $a_1 = 5$ $a_2 = a_1 + 3 = 5 + 3 = 8$ $a_3 = a_2 + 3 = 8 + 3 = 11$ $a_4 = a_3 + 3 = 11 + 3 = 14$ $a_5 = a_4 + 3 = 14 + 3 = 17$

Answer: The first five terms of the sequence are: $5, 8, 11, 14, 17, \dots$

---

2. Geometric Sequence:

Recursive equation: $b_1 = 2 \quad \text{and} \quad b_n = 2b_{n-1} \text{ for } n > 1$

Step-by-step solution:

• Given: $b_1 = 2$ and the recursive formula $b_n = 2b_{n-1}$.
• To find the first few terms of the sequence, we start from $b_1$.

Finding the terms: $b_1 = 2$ $b_2 = 2b_1 = 2 \times 2 = 4$ $b_3 = 2b_2 = 2 \times 4 = 8$ $b_4 = 2b_3 = 2 \times 8 = 16$ $b_5 = 2b_4 = 2 \times 16 = 32$

Answer: The first five terms of the sequence are: $2, 4, 8, 16, 32, \dots$

---

3. Fibonacci Sequence:

Recursive equation: $c_1 = 1, \quad c_2 = 1 \quad \text{and} \quad c_n = c_{n-1} + c_{n-2} \text{ for } n > 2$

Step-by-step solution:

• Given: $c_1 = 1$, $c_2 = 1$, and the recursive formula $c_n = c_{n-1} + c_{n-2}$.
• To find the first few terms of the sequence, we start from $c_1$ and $c_2$.

Finding the terms: $c_1 = 1, \quad c_2 = 1$ $c_3 = c_2 + c_1 = 1 + 1 = 2$ $c_4 = c_3 + c_2 = 2 + 1 = 3$ $c_5 = c_4 + c_3 = 3 + 2 = 5$ $c_6 = c_5 + c_4 = 5 + 3 = 8$

Answer: The first five terms of the sequence are: $1, 1, 2, 3, 5, 8, \dots$

---

These are three common types of recursive sequences. Would you like to see more examples or further explanation?

Related Questions:

1. What happens if the common difference of an arithmetic sequence is negative?
2. How does the geometric sequence behave if the common ratio is a fraction between 0 and 1?
3. Can we derive a non-recursive formula for the Fibonacci sequence?
4. How can we prove that the sum of an arithmetic series is always a fixed value?
5. How can the Fibonacci sequence be used in nature or real-world scenarios?

Tip: To work with recursive sequences effectively, always start with the initial terms and apply the recursive rule step by step. It helps to also recognize patterns that might emerge from the sequence!