What is the pattern in the sequence 2, 4, 8, 16?

The pattern in the sequence 2, 4, 8, 16 is that each term is double the previous term.

This sequence is an example of a geometric progression, where each term is obtained by multiplying the previous term by a constant factor. In this case, the constant factor is 2. Starting with the first term, 2, you multiply by 2 to get the second term, 4. Then, you multiply 4 by 2 to get the third term, 8, and so on. This pattern continues indefinitely, with each term being twice the value of the term before it.

To analyse this further, let's denote the first term of the sequence as \( a \). Here, \( a = 2 \). The common ratio, which is the factor by which we multiply each term to get the next term, is \( r = 2 \). The general formula for the \( n \)-th term of a geometric sequence is given by \( a \times r^{(n-1)} \). For this sequence, the \( n \)-th term can be written as \( 2 \times 2^{(n-1)} \). For example, the 4th term is \( 2 \times 2^{(4-1)} = 2 \times 2^3 = 2 \times 8 = 16 \).

Understanding this pattern helps in predicting future terms of the sequence without having to list all the previous terms. For instance, the 5th term would be \( 2 \times 2^{(5-1)} = 2 \times 2^4 = 2 \times 16 = 32 \). This method of using the general formula is particularly useful for sequences with a large number of terms.