4.3 Square, Cube & Figurate Number Sequences

Square, Cube & Figurate Number Sequences

Core Concept: Non-linear sequences based on powers and figurate numbers cannot be solved efficiently by raw trial-and-error under test conditions. Memorizing key numerical benchmarks—perfect squares through $25^2$, perfect cubes through $12^3$, and powers of 2 up to $2^{10}$—transforms complex polynomial sequences into instantaneous pattern-recognition matches.

Series built on squares, cubes and figurate numbers are a staple of numerical reasoning tests. They reward preparation because untrained candidates can take 60 to 90 seconds attempting three tiers of subtractions, whereas a student with fluent benchmark recall recognizes the numbers in under 10 seconds.

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Mental Benchmarks: Non-Negotiable Memorisation Tables

Without a calculator, your mental memory must act as your reference lookup table. Commit every value in the following table to instant visual recall.

Core Power Benchmarks Reference Table

High-Speed Squaring Shortcuts

• Numbers Ending in 5: To square any two-digit number ending in $5$ ($N5$): Multiply the tens digit $N$ by $(N + 1)$ and append $25$.
  • $35^2 \implies 3 \times 4 = 12 \implies 1225$
  • $65^2 \implies 6 \times 7 = 42 \implies 4225$
  • $85^2 \implies 8 \times 9 = 72 \implies 7225$
• Difference of Two Squares: $a^2 - b^2 = (a - b)(a + b)$. Use this to compute products of numbers equidistant from a round benchmark:
  • $29 \times 31 = (30 - 1)(30 + 1) = 30^2 - 1^2 = 900 - 1 = 899$
  • $48 \times 52 = (50 - 2)(50 + 2) = 50^2 - 2^2 = 2500 - 4 = 2496$

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Square Sequences & Odd Difference Signatures

The fundamental algebraic theorem connecting squares and differences is:

$(n+1)^2 - n^2 = (n^2 + 2n + 1) - n^2 = 2n + 1$

Because $2n + 1$ always generates an odd integer, the difference between consecutive square numbers is always the sequence of consecutive odd numbers:

Squares:      1       4       9       16      25      36      49      64
               \     / \     / \     /  \    /  \    /  \    /  \    /
First Diff:       3       5       7       9      11      13      15   (Consecutive Odds)
                   \     / \     / \     / \    /  \    /  \    /
Second Diff:          2       2       2       2       2       2       (Constant = 2)

Whenever your first differences produce the sequence $3, 5, 7, 9, 11$ (or any shifted subset of odd numbers, such as $13, 15, 17, 19$), you are dealing with a quadratic sequence governed by $n^2$.

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Disguised Square-Offset Sequences ($n^2 \pm k$ and $n^2 \pm n$)

Examiners rarely present bare perfect squares ($1, 4, 9, 16, \dots$). Instead, they add or subtract an offset to test your benchmark pattern recognition.

1. Constant Square Offsets ($n^2 \pm k$)

• Offset $n^2 - 1$: $0, 3, 8, 15, 24, 35, 48, 63, 80, \dots$ Notice these numbers are one less than $1, 4, 9, 16, 25, 36, 49, 64, 81$.
• Offset $n^2 + 1$: $2, 5, 10, 17, 26, 37, 50, 65, 82, \dots$
• Offset $n^2 + 2$: $3, 6, 11, 18, 27, 38, 51, 66, 83, \dots$

Worked Example: Spotting a Shifted Square Offset

$\text{Find the missing term: } 26,\ 37,\ 50,\ 65,\ 82,\ ?$

• Method A (Benchmark Recognition): Examine each term against your benchmark table: $26 = 25 + 1 = 5^2 + 1$ $37 = 36 + 1 = 6^2 + 1$ $50 = 49 + 1 = 7^2 + 1$ $65 = 64 + 1 = 8^2 + 1$ $82 = 81 + 1 = 9^2 + 1$ The next term is $10^2 + 1 = 100 + 1 = 101$. Solved in under 8 seconds!
• Method B (Difference Verification): $37 - 26 = 11$; $50 - 37 = 13$; $65 - 50 = 15$; $82 - 65 = 17$. The differences are consecutive odd numbers ($11, 13, 15, 17$). Next difference is $19$. $82 + 19 = 101$.

2. Variable Square Offsets (Pronic Numbers: $n^2 \pm n$)

When the offset is not a constant, but equals the index $n$ itself, the resulting numbers are known as pronic numbers (or rectangular numbers), representing products of consecutive integers $n(n+1)$ or $n(n-1)$:

$n^2 + n = n(n+1) \implies 2, 6, 12, 20, 30, 42, 56, 72, 90, 110, \dots$ $n^2 - n = n(n-1) \implies 0, 2, 6, 12, 20, 30, 42, 56, 72, 90, \dots$

First differences for pronic numbers are consecutive even integers: $+4, +6, +8, +10, +12, +14, \dots$

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Cube Sequences & Third-Order Differences ($n^3 \pm k$ and $n^3 \pm n$)

The difference between consecutive cube numbers expands rapidly:

$(n+1)^3 - n^3 = 3n^2 + 3n + 1$

Cubes:        1       8       27      64      125     216     343
               \     / \     /  \    /  \    /   \   /   \   /
First Diff:       7       19      37      61      91     127
                   \     /  \    /  \    /  \    /  \   /
Second Diff:          12      18      24      30      36     (Arithmetic: +6)
                       \     /  \    /  \    /  \    /
Third Diff:                6       6       6       6         (Constant = 6 = 3!)

A Classic Cube-Offset Sequence: $n^3 - n$

A well-known non-linear pattern in reasoning puzzles is:

$0,\ 6,\ 24,\ 60,\ 120,\ 210,\ 336,\ 504, \dots$

This sequence has two equivalent definitions that every candidate should memorize:

1. Cube minus Index: $n^3 - n$ $1^3 - 1 = 0$ $2^3 - 2 = 6$ $3^3 - 3 = 24$ $4^3 - 4 = 60$ $5^3 - 5 = 120$ $6^3 - 6 = 210$ $7^3 - 7 = 343 - 7 = 336$
2. Product of Three Consecutive Integers: $n^3 - n = n(n^2 - 1) = (n - 1) \cdot n \cdot (n + 1)$
   • $0 \times 1 \times 2 = 0$
   • $1 \times 2 \times 3 = 6$
   • $2 \times 3 \times 4 = 24$
   • $3 \times 4 \times 5 = 60$
   • $4 \times 5 \times 6 = 120$
   • $5 \times 6 \times 7 = 210$
   • $6 \times 7 \times 8 = 336$

If you memorize this exact sequence, an item that takes other students 75 seconds of painful cubic difference calculation becomes a 5-second automatic recall question.

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Triangular Numbers & Figurate Geometry

A triangular number counts objects arranged in an equilateral triangle. The $n$-th triangular number ($T_n$) is the sum of the first $n$ positive integers:

$T_n = \sum_{i=1}^n i = 1 + 2 + 3 + \dots + n = \frac{n(n+1)}{2}$

$\text{Triangular Numbers: } 1,\ 3,\ 6,\ 10,\ 15,\ 21,\ 28,\ 36,\ 45,\ 55,\ 66,\ 78,\ 91,\ 105, \dots$

Pattern:     1          3              6                  10
Visual:      ●          ●              ●                   ●
                       ● ●            ● ●                 ● ●
                                     ● ● ●               ● ● ●
                                                        ● ● ● ●
Diff:            +2             +3               +4

Essential Mathematical Properties of Triangular Numbers

1. First Differences: The difference between consecutive triangular numbers increases by exactly $1$ at each step ($+2, +3, +4, +5, +6, \dots$). The second difference is constant at $\Delta_2 = 1$.
2. Sum of Two Consecutive Triangular Numbers is a Square: $T_{n-1} + T_n = \frac{(n-1)n}{2} + \frac{n(n+1)}{2} = \frac{n^2 - n + n^2 + n}{2} = n^2$
   • $1 + 3 = 4 = 2^2$
   • $3 + 6 = 9 = 3^2$
   • $6 + 10 = 16 = 4^2$
   • $10 + 15 = 25 = 5^2$
   • $15 + 21 = 36 = 6^2$
   • $21 + 28 = 49 = 7^2$
3. Handshake / Round-Robin Tournament Formula: The number of distinct pairings among $n$ people or teams is $T_{n-1} = \frac{n(n-1)}{2}$.

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Figurate & Power Progressions at a Glance

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High-Yield Traps in Figurate & Power Patterns

1. The Brute-Force Difference Trap on Cubes

Calculating third-order differences on an item like $0, 6, 24, 60, 120, 210$ requires computing $5$ first differences, $4$ second differences, and $3$ third differences—consuming up to $60$ seconds of intense mental arithmetic. If terms pass $100$ quickly and third differences are involved, immediately check whether the numbers cluster near perfect cubes ($n^3$) or cube offsets ($n^3 \pm n$).

2. Base Index Misalignment ($n = 0$ vs $n = 1$)

When working with sequences like $0, 3, 8, 15, 24$, candidates often get confused about whether $0$ corresponds to $n = 1$ ($1^2 - 1 = 0$) or $n = 0$. Always anchor your index numbering to the specific terms. In $n^2 - 1$, the first term ($n = 1$) is $1^2 - 1 = 0$, the second term ($n = 2$) is $2^2 - 1 = 3$, and so forth.

3. Confusing Pronic Numbers with Triangular Numbers

Pronic numbers ($n(n+1)$) and triangular numbers ($\frac{n(n+1)}{2}$) share the exact same structural roots, but pronic numbers are exactly twice the corresponding triangular numbers ($2 = 2 \times 1$, $6 = 2 \times 3$, $12 = 2 \times 6$, $20 = 2 \times 10$). If the differences are consecutive even numbers ($+4, +6, +8$), it is pronic; if the differences are consecutive integers ($+2, +3, +4$), it is triangular.