The sequence $2, 4, 8, 16$ illustrates a clear pattern: each term is double the preceding term.

This sequence represents a geometric progression, where each term is generated by multiplying the previous term by a constant factor. In this case, the constant factor is $2$. Starting with the first term, $2$, we multiply by $2$ to obtain the second term, $4$. Continuing this process, we multiply $4$ by $2$ to arrive at the third term, $8$, and so forth. This doubling pattern continues indefinitely, with each term consistently being twice the value of its predecessor.

To delve deeper into this progression, let’s denote the first term of the sequence as $a$. Here, we have $a = 2$. The common ratio, which is the factor used to obtain each successive term, is $r = 2$. The general formula for the $n$-th term of a geometric sequence is given by

For this specific sequence, the $n$-th term can be expressed as

For example, to calculate the 4th term:

Understanding this pattern enables us to predict future terms of the sequence without listing all the preceding terms. For instance, the 5th term can be calculated as follows:

Utilizing the general formula is especially advantageous for sequences with a large number of terms, as it allows for quick calculations and predictions.