7.1 Arithmetic, Geometric & Two-Stage Number Sequences

7.1 Arithmetic, Geometric & Two-Stage Number Sequences

This chapter offers optional number-pattern practice alongside numerical data interpretation. Publicjobs describes logical thinking separately from numerical reasoning; do not assume these patterns form part of every numerical assessment or have a fixed question time.

A sequence is an ordered list; in formal mathematics, a series is a sum of terms. Pattern exercises often use “number series” informally for a sequence. A finite prefix can fit more than one rule. The exercises here specify a rule family or recurrence so that continuation is conditional on that model, rather than proof that no other continuation exists.

---

Linear Arithmetic Progressions (Constant First Difference $\Delta_1$)

An arithmetic progression is the most fundamental sequence type. Each successive term is generated by adding or subtracting a fixed constant, known as the common difference ($d$):

$T_n = a + (n - 1)d$

Where $a$ is the initial term and $d$ is the common difference ($T_n - T_{n-1} = d$).

Characteristics and Variants

• Positive and Negative Steps: Common differences can be positive integers ($+6, +14$), negative integers ($-5, -11$), or fractional and decimal values ($+2.5, -0.75$).
• Diagnostic Rule: If the first difference row ($\Delta_1$) between consecutive terms is strictly identical through the observed terms, those terms fit an arithmetic model. The rule or problem assumptions determine its continuation.
• Administrative Context: A hypothetical model with fixed annual retirements or constant weekly additions can use an arithmetic sequence, provided the rule is explicitly stated.

$\text{Example: } 17, 24, 31, 38, 45, 52 \implies \Delta_1 = +7 \quad (\text{Constant})$

---

Geometric Progressions (Common Ratio $r$)

A geometric sequence advances not by constant addition, but through continuous multiplication or division by a fixed factor, termed the common ratio ($r$):

$T_n = a \cdot r^{n-1}$

The recurrence is $T_n=rT_{n-1}$ for $n\ge2$. When the preceding term is nonzero, its ratio can be found from: $r = \frac{T_n}{T_{n-1}}$.

Growth Profiles and Alternating Signs

• Expansive Growth ($|r| > 1$): For a nonzero initial term, absolute values grow geometrically (e.g., $\times 2, \times 3, \times 1.5$).
• Contractive Decay ($0 < |r| < 1$): Terms shrink toward zero (e.g., $\times 0.5, \times \frac{2}{3}$).
• Oscillating Signs ($r < 0$): With a nonzero seed and negative ratio, terms alternate between positive and negative values (e.g., $3, -6, 12, -24, 48 \implies r = -2$).
• Diagnostic Rule: When the differences between adjacent terms ($\Delta_1$) increase or decrease in direct proportion to the terms themselves, evaluate the ratio $\frac{T_{n}}{T_{n-1}}$. If the quotient remains invariant, the sequence is geometric.

$\text{Example: } 8, 20, 50, 125, 312.5 \implies r = \frac{20}{8} = \frac{50}{20} = 2.5$

---

Power-Based Sequences: Squares, Cubes, and Polynomial Offsets

Psychometric item writers frequently construct sequences based on exponents, as raw differences can appear confusing without recognition of the underlying powers.

Benchmark Powers to Memorize

Benchmark powers provide useful reference values:

• Benchmark Squares ($n^2$): $1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400$.
• Benchmark Cubes ($n^3$): $1, 8, 27, 64, 125, 216, 343, 512, 729, 1{,}000$.

Constant-Offset Power Series ($n^2 \pm k$ and $n^3 \pm k$)

In these sequences, a uniform integer $k$ is added to or subtracted from each successive power:

• $n^2 + 1$: $2, 5, 10, 17, 26, 37, 50, \dots$
• $n^2 - 2$: $-1, 2, 7, 14, 23, 34, 47, \dots$
• $n^3 - 1$: $0, 7, 26, 63, 124, 215, 342, \dots$

Index-Variable Power Series ($n^2 \pm n$ and $n^3 \pm n$)

Here, the offset is not fixed, but equals the index $n$ of the term itself:

• $n^2 - n$ (or $n(n-1)$): $0, 2, 6, 12, 20, 30, 42, 56, \dots$ (commonly called pronic numbers).
• $n^2 + n$ (or $n(n+1)$): $2, 6, 12, 20, 30, 42, 56, 72, \dots$
• $n^3 - n$: $0, 6, 24, 60, 120, 210, 336, \dots$

---

Two-Stage / Quadratic Sequences (Differences of Differences $\Delta_2$)

When the first differences between consecutive terms ($\Delta_1$) are not constant, candidates must examine the second difference row ($\Delta_2$). At consecutive integer indices with step 1, a quadratic formula has constant nonzero second differences. An observed finite prefix with that signature fits a quadratic continuation; zero second differences instead fit a linear or constant sequence.

Mathematical Formulation

A quadratic sequence conforms to the second-order polynomial equation:

$T_n = an^2 + bn + c$

• The first difference row ($\Delta_1$) forms an arithmetic progression with common difference $2a$.
• The second difference row ($\Delta_2$) is constant and equal to $2a$.

Systematic Diagnostic Walkthrough

Consider the sequence: $5, 12, 23, 38, 57, ?$

1. Compute First Differences ($\Delta_1$): $12 - 5 = 7$ $23 - 12 = 11$ $38 - 23 = 15$ $57 - 38 = 19$ The first differences are $7, 11, 15, 19$. They are not constant, ruling out a simple linear arithmetic progression.

2. Compute Second Differences ($\Delta_2$): $11 - 7 = 4$ $15 - 11 = 4$ $19 - 15 = 4$ The second differences are constant at $\Delta_2 = +4$.

3. Project the Next Step:

   • Extend the first difference row: $19 + 4 = 23$.
   • Add this extended difference to the final known term: $57 + 23 = 80$. Under the constant-second-difference rule, the next term is 80.

---

The Systematic 4-Step Sequence Triage Protocol

Under timed psychometric assessment conditions, avoid random guessing. Execute this four-step triage procedure on your scratch sheet:

[Step 1: Growth Velocity Inspection]
   │
   ├── Steady / Slow Growth ─────────► Check Linear Arithmetic (+d / -d)
   ├── Rapid / Accelerating Growth ──► Check Quadratic (Delta_2) or Squares (n^2)
   └── Explosive Growth ─────────────► Check Geometric (* r) or Cubes (n^3)
   │
[Step 2: Construct Delta_1 (First Differences)]
   │
   ├── Delta_1 is Constant ──────────► Linear Arithmetic: T_next = T_last + d
   └── Delta_1 is Non-Constant ──────► Proceed to Step 3
   │
[Step 3: Construct Delta_2 (Second Differences)]
   │
   ├── Delta_2 is Constant ──────────► Quadratic: Extend Delta_1, then compute T_next
   └── Delta_2 is Non-Constant ──────► Proceed to Step 4
   │
[Step 4: Check Multiplicative Ratios & Power Offsets]
   ├── Test Constant Ratio (T_n / T_{n-1} = r)
   └── Test Proximity to Benchmark Powers (n^2 +/- k, n^3 +/- k)

---

Sequence Taxonomy & Diagnostic Rules Reference

The following reference table outlines the primary single-rule sequence categories, their mathematical formulations, and their difference signatures: