In mathematics, sequences can be defined using different types of formulas, primarily general formulas and recursive formulas. A general formula provides a direct way to calculate the nth term of a sequence based on its position, n. For example, the formula \( a_n = 2n \) allows you to find the first five terms of the sequence by substituting values of n (1, 2, 3, 4, 5) to yield the outputs 2, 4, 6, 8, and 10.

On the other hand, a recursive formula defines each term based on the previous term(s) in the sequence. For instance, a recursive formula might be expressed as \( a_n = a_{n-1} + 2 \), where \( a_{n-1} \) represents the previous term. To find the terms of the sequence using this formula, you start with the first term, \( a_1 \), and use it to calculate subsequent terms. If \( a_1 = 2 \), then:

1. \( a_2 = a_1 + 2 = 2 + 2 = 4 \)

2. \( a_3 = a_2 + 2 = 4 + 2 = 6 \)

3. \( a_4 = a_3 + 2 = 6 + 2 = 8 \)

4. \( a_5 = a_4 + 2 = 8 + 2 = 10 \)

This results in the same sequence as derived from the general formula. The key distinction is that while the general formula requires knowledge of n to compute the nth term, the recursive formula relies on the previous term to find the next one.

Recursive formulas can be particularly useful when the pattern of the sequence is evident, allowing for easier calculations without needing to derive a general formula. For example, if given a recursive formula \( a_n = 2 \cdot a_{n-1} + 3 \) with \( a_1 = 1 \), the terms can be calculated as follows:

1. \( a_2 = 2 \cdot a_1 + 3 = 2 \cdot 1 + 3 = 5 \)

2. \( a_3 = 2 \cdot a_2 + 3 = 2 \cdot 5 + 3 = 13 \)

3. \( a_4 = 2 \cdot a_3 + 3 = 2 \cdot 13 + 3 = 29 \)

In this case, the sequence generated is 1, 5, 13, 29. Finding a general formula for such sequences can be complex, making recursive formulas advantageous for determining a few terms quickly.

Another example could involve a recursive formula defined as \( a_n = n \cdot a_{n-1} \) with \( a_1 = 1 \). The terms can be calculated as follows:

1. \( a_2 = 2 \cdot a_1 = 2 \cdot 1 = 2 \)

2. \( a_3 = 3 \cdot a_2 = 3 \cdot 2 = 6 \)

3. \( a_4 = 4 \cdot a_3 = 4 \cdot 6 = 24 \)

This results in the sequence 1, 2, 6, 24, showcasing how recursive formulas can effectively generate terms based on previous values. Understanding both types of formulas enhances your ability to work with sequences in various mathematical contexts.