Square, Cube and Prime Based Series

The series that stump most candidates are built on squares, cubes and primes hidden behind small adjustments. If your squares up to 20 and cubes up to 12 are instant, these become some of the fastest questions in the paper; if they are not, you will waste a minute and still miss the pattern. This is where memorised tables pay off directly, and it is the single strongest argument for spending revision time on rote recall of squares, cubes and primes rather than on yet more mixed practice sets.

Square-Based Series

Terms sit at or near consecutive squares. The plain form is 1, 4, 9, 16, 25; the disguised forms add or subtract a small constant, such as 2, 5, 10, 17, 26 which is n² + 1. When numbers look 'almost square', compare each with the nearest perfect square and read off the adjustment.

n² + k pattern: 2, 5, 10, 17, 26 → 1²+1, 2²+1, 3²+1, 4²+1, 5²+1

Cube-Based Series

Terms follow consecutive cubes, sometimes with a shift. A row like 2, 9, 28, 65 is 1³+1, 2³+1, 3³+1, 4³+1. Because cubes grow fast, a cube series is easy to confuse with a geometric one, so if the ratio is not constant, check against 1, 8, 27, 64, 125 before giving up.

Prime-Based Series

Some series step through prime numbers — 2, 3, 5, 7, 11, 13 — or add successive primes as the gap: +2, +3, +5, +7. Keeping the first several primes ready in memory lets you spot these instantly, where a candidate hunting for an arithmetic rule will simply never find one. A quick glance for the tell-tale prime sequence at the start of a series often settles the whole question in one move.

Solved Examples

Example 1: 3, 6, 11, 18, ? is n² + 2: 1²+2, 2²+2, 3²+2, 4²+2, so next is 5²+2 = 27.

Example 2: 1, 8, 27, 64, ? are cubes 1³ to 4³, so the next is 5³ = 125.

Example 3: 4, 6, 10, 16, ? adds successive primes 2, 4... use gaps +2, +4, +6 → next +8 wait — use primes: 2, 3, 5 gaps... 4+2=6, 6+... take 4, 6, 9, 13 pattern; for 2, 5, 10, 17 style, answer follows n²+1 → 26.

✗ Forcing a ratio on 2, 9, 28, 65  |  ✓ Recognising 1³+1, 2³+1, 3³+1, 4³+1

Combined and Multiplied-Power Series

Tougher papers combine two ideas, such as squares of consecutive numbers each multiplied by a constant, or a series where the term is n² + n. A row like 2, 6, 12, 20, 30 is not obviously square, yet it equals 1×2, 2×3, 3×4, 4×5 — the product of consecutive integers, which also equals n² + n. When simple squares and cubes do not fit but the growth still feels 'quadratic' rather than geometric, test products of consecutive numbers. Keeping this one extra pattern in your toolkit covers almost every remaining power-based series the examiner can build.