Number Sequences

Arithmetic Sequences (Add or Subtract)

A sequence is an ordered list of numbers that follow a rule. The same amount is added or subtracted each time — this is called the common difference.

• 3, 7, 11, 15, 19   (+4 each time)
• 100, 93, 86, 79, 72   (−7 each time)

Geometric Sequences (Multiply or Divide)

The same number is multiplied or divided each time — this is the common ratio.

• 2, 6, 18, 54, 162   (×3 each time)
• 256, 64, 16, 4, 1   (÷4 each time)

Other Common Patterns

• Square numbers: 1, 4, 9, 16, 25, 36   (n²)
• Fibonacci-type: 1, 1, 2, 3, 5, 8, 13   (add the two previous terms)
• Increasing difference: 1, 2, 4, 7, 11, 16   (differences are +1, +2, +3, +4...)
• Alternating: 2, 10, 4, 8, 6, 6, 8, 4   (two interleaved sequences)

How to Find the Rule & Key Tips

1. Calculate differences between consecutive terms — is the difference constant?
2. If not, check the ratio (is each term multiplied by the same number?).
3. If still unclear, find the second differences (differences of the differences).
4. Check odd-position and even-position terms separately for alternating sequences.

Key Tips

• Always write down the differences between terms before guessing the rule
• For increasing differences, the second difference reveals a quadratic pattern
• Negative sequences work the same way — extend the pattern past zero if needed
• If the sequence has a missing term in the middle, work forward and backward to check