Number Series
Find the missing term in number sequences
Number series questions ask you to identify the rule connecting successive terms and then use it to find a missing or next term. The rule may involve a constant difference, a constant ratio, powers, a recurrence, alternating positions or successive differences. A finite list can fit many invented rules, so prefer the simplest consistent rule that explains every term and fits the answer choices.
Key formulas and rules
Key concepts
Arithmetic progressions
Subtract consecutive terms. If the same difference appears throughout, the series is arithmetic. The difference may be positive or negative, and it may be fractional. For example, 18, 13, 8, 3 has d = -5, so the next term is -2.
Geometric progressions
Divide each term by the preceding term when division is defined. A constant ratio identifies a geometric pattern, such as 3, 6, 12, 24 with r = 2. A decreasing series may have a fractional ratio, such as 81, 27, 9, 3 with r = 1/3.
Powers and familiar sequences
Check common structured families after testing simple differences and ratios. Useful candidates include squares, cubes, primes, multiples, triangular numbers and factorials. Write the index beside each term, because a pattern such as n squared plus 1 becomes easier to see when the position is explicit.
Difference tables
If first differences are not constant, calculate differences again. A constant second difference often indicates a quadratic rule; a constant third difference can indicate a cubic rule. Use this method only after checking simpler patterns, and verify the derived next difference before selecting an answer.