In mathematics, sequences can be categorized into different types based on the relationship between their terms. Two primary types are arithmetic sequences and geometric sequences. An arithmetic sequence, such as 3, 6, 9, 12, has a constant difference between consecutive terms, known as the common difference (denoted as d). For example, in this sequence, d equals 3, as each term increases by 3.

In contrast, a geometric sequence, like 3, 9, 27, 81, has a constant ratio between consecutive terms, referred to as the common ratio (denoted as r). For instance, moving from 3 to 9 involves multiplying by 3, and this pattern continues with each subsequent term. Thus, in this case, r is also equal to 3.

To express the relationship in a geometric sequence, a recursive formula can be established. This formula allows you to find the next term based on the previous term. The general structure for a geometric sequence is given by:

A_n = A_n-1 × r

Here, A_n represents the new term, A_n-1 is the previous term, and r is the common ratio. For example, if we consider the sequence 5, 20, 80, and 320, we can determine the common ratio by dividing any two consecutive terms. From 5 to 20, we multiply by 4; from 20 to 80, we again multiply by 4; and from 80 to 320, we multiply by 4 once more. Therefore, the common ratio r is 4.

Using this common ratio, we can write the recursive formula for this geometric sequence as:

A_n = A_n-1 × 4

Additionally, it is essential to specify the first term of the sequence, which in this case is:

A₁ = 5

Thus, to find subsequent terms, one would simply multiply the previous term by 4. This method highlights the rapid growth characteristic of geometric sequences, as they tend to increase exponentially compared to the linear growth of arithmetic sequences.