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 index, n. For example, if we have a sequence defined by the formula \( a_n = 2n \), we can easily find the first five terms by substituting n with 1, 2, 3, 4, and 5, yielding the terms 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 \( a_n = a_{n-1} + 2 \). Here, \( a_{n-1} \) represents the previous term, and to find the next term, you simply add 2 to it. Starting with the first term \( a_1 = 2 \), you can calculate subsequent terms: \( a_2 = a_1 + 2 = 4 \), \( a_3 = a_2 + 2 = 6 \), and so forth, resulting in the same sequence of 2, 4, 6, 8, and 10.

The key distinction between general and recursive formulas lies in their approach: general formulas require knowledge of n to compute the nth term, while recursive formulas rely on the previous term to derive the next one. This makes recursive formulas particularly useful when only a few terms need to be calculated, as they can simplify the process without needing to derive a general formula.

For example, consider a recursive formula defined as \( a_n = 2 \cdot a_{n-1} + 3 \) with \( a_1 = 1 \). To find the second term, you would calculate \( a_2 = 2 \cdot a_1 + 3 = 2 \cdot 1 + 3 = 5 \). Continuing this process, the third term would be \( a_3 = 2 \cdot a_2 + 3 = 2 \cdot 5 + 3 = 13 \), and the fourth term would be \( a_4 = 2 \cdot a_3 + 3 = 2 \cdot 13 + 3 = 29 \).

Another example could involve a recursive formula like \( a_n = n \cdot a_{n-1} \) with \( a_1 = 1 \). To find the second term, you would compute \( a_2 = 2 \cdot a_1 = 2 \cdot 1 = 2 \). For the third term, \( a_3 = 3 \cdot a_2 = 3 \cdot 2 = 6 \), and for the fourth term, \( a_4 = 4 \cdot a_3 = 4 \cdot 6 = 24 \).

In summary, both general and recursive formulas are valuable tools for defining sequences, each serving different purposes depending on the context and requirements of the problem at hand. Understanding how to utilize both types of formulas enhances your ability to work with sequences effectively.