Patterns

Pattern problems require identifying the underlying rule of a sequence. There is no single trick, but it's useful to look for common patterns such as:

• Arithmetic/Geometric progressions: Check for a common difference or ratio.
• Powers: Look for squares, cubes, etc., possibly with an offset (e.g., $n^3 - 1$).
• Combinations of patterns: The sequence might be a sum of other sequences (e.g., $n^2 + (n-1)^2$).

For example, in the sequence 0, 7, 26, 63, ..., each term is one less than a cube: $1^3-1, 2^3-1, 3^3-1, 4^3-1$. The next term is $5^3-1=124$.