Number Series

A number series is a sequence of numbers arranged by a hidden rule. Your task is to find that rule and use it to identify the next number (or the missing number in the middle). Common rules: add a fixed number (2, 4, 6, 8 — add 2), multiply (2, 4, 8, 16 — multiply by 2), squares (1, 4, 9, 16, 25 — perfect squares), cubes (1, 8, 27, 64), or alternating operations (add 2 then add 3 alternately).

1. Calculate the difference between consecutive terms: write the gaps above the arrows.
2. If differences are equal — rule is "add constant."
3. If differences themselves increase by a fixed amount — rule is "adding increasing values" (second-order arithmetic).
4. If each term is multiplied by the same number — geometric series.
5. If none of the above, check for squares, cubes, primes, or alternating two rules.

Example 1: 3, 6, 12, 24, ?

• Gaps: 6-3=3, 12-6=6, 24-12=12. Each gap doubles.
• Alternatively: each term is multiplied by 2.
• Next term: 24 x 2 = 48.
• Answer: 48.

Example 2: 1, 4, 9, 16, 25, ?

• Terms are 1 squared, 2 squared, 3 squared, 4 squared, 5 squared.
• Next: 6 squared = 36.
• Answer: 36.

• Stopping at the first differences without checking if those differences also form a pattern.
• Mistaking a geometric series for an arithmetic one when numbers grow quickly — check ratios, not just differences.
• Missing series where the rule alternates: 2, 5, 3, 6, 4, 7 — odd positions decrease? No — odd positions: 2,3,4 (+1); even positions: 5,6,7 (+1). Two interleaved series.

1. 5, 10, 20, 40, ? — find the next number.
2. 2, 5, 10, 17, 26, ? — find the next number (hint: look at the differences).