The sequence $2, 6, 12, 20$ exhibits a clear pattern: each term is derived from the product of two consecutive integers, with an additional $2$ added.

To analyze this sequence more thoroughly, let’s examine how each term is constructed. The first term is $2$, which can be expressed as $1 \times 2$. The second term is $6$, which corresponds to $2 \times 3$. The third term, $12$, can be represented as $3 \times 4$, and the fourth term, $20$, is $4 \times 5$.

Notice that each term in the sequence can be formulated as the product of two consecutive integers. Specifically, the $n$th term can be represented as:

For instance, the first term is calculated as:

The second term is:

Continuing this pattern, the third term becomes:

And similarly, the fourth term is:

This pattern can be generalized for any term in the sequence. To find the $n$th term, simply multiply $n$ by $(n + 1)$. This approach allows you to efficiently determine any term in the sequence without needing to enumerate all the preceding terms.

Understanding this pattern is essential for recognizing how sequences are formed and manipulated, which is a crucial skill in GCSE Mathematics.