4.1 Arithmetic, Geometric & Two-Tier Differences

Arithmetic, Geometric & Two-Tier Differences

Core Concept: Number sequences on the Edutest Numerical Reasoning subtest test structural pattern recognition under severe time pressure. The key to working through 40 to 60 questions in 30 minutes without a calculator is not intuitive guessing, but applying a fixed triage order within the first 15 seconds of reading the item.

Edutest describes Numerical Reasoning as measuring the ability to think and reason using numbers, including series, matrices, arithmetical reasoning and deduction. It evaluates mathematical reasoning rather than curriculum recall. While mathematics achievement tests assess formulas, geometry theorems, and multi-step word problems, numerical reasoning evaluates how rapidly and flexibly your brain identifies hidden numerical relationships among abstract data.

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Exam Architecture & The Time Budget

Edutest's Numerical Reasoning test lasts 30 minutes, and Edutest states that actual tests contain between 40 and 60 questions. That gives you an average of only 30 to 45 seconds per question.

Furthermore, calculators are strictly forbidden. Every calculation—from multi-digit addition to fractional division—must be performed mentally or with brief written working, because Edutest does not permit separate working out paper. Edutest does not publish how many items are series questions, but series are the first skill it names for this test. Because there is no penalty for incorrect answers, maintaining momentum is essential: a candidate who spends 90 seconds stuck on a single complex sequence severely handicaps their overall percentile rank.

[30-Minute Total Time Window]
  ├── Total Questions: 40 to 60 items (Edutest's stated range)
  ├── Average Time per Item: 30–45 seconds
  ├── Target Time on Sequences: 20–30 seconds (to bank time for logic word puzzles)
  └── Triage Rule: If pattern is undetected at 20 seconds, eliminate extremes and guess.

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Arithmetic Progressions (First-Order Linear Series)

An arithmetic progression is a sequence in which each term after the first is generated by adding a fixed constant value, known as the common difference ($d$).

$\text{General Term: } T_n = a + (n-1)d$

Where:

• $a$ is the initial term ($T_1$)
• $n$ is the term index position ($1, 2, 3, \dots$)
• $d$ is the common difference ($T_n - T_{n-1}$)

1. Integer Step Variations

Arithmetic sequences on scholarship exams rarely feature trivial small-number additions like $+3$. Instead, examiners test your agility across three domains:

• Large Positive Steps: Involving mental additions across decade boundaries, such as $+17$ or $+23$ (e.g., $14, 31, 48, 65, 82, \dots$).
• Negative Steps (Decelerating Series): Descending across zero into negative integers, such as $-14$ (e.g., $29, 15, 1, -13, -27, \dots$).

2. Fractional and Decimal Steps

Examiners frequently disguise basic arithmetic progressions by expressing terms as mixed numerals with unlike denominators. To uncover the common difference, convert all fractions to a common denominator immediately.

Worked Example: Fractional Arithmetic Progression

$\text{Find the missing term: } 1\frac{1}{4},\ 2\frac{7}{12},\ 3\frac{11}{12},\ 5\frac{1}{4},\ ?$

• Step 1 (Convert to common denominator 12): $T_1 = 1\frac{3}{12} = \frac{15}{12}$ $T_2 = 2\frac{7}{12} = \frac{31}{12}$ $T_3 = 3\frac{11}{12} = \frac{47}{12}$ $T_4 = 5\frac{3}{12} = \frac{63}{12}$
• Step 2 (Calculate first differences $\Delta_1$): $T_2 - T_1 = \frac{31}{12} - \frac{15}{12} = \frac{16}{12} = \frac{4}{3} = 1\frac{1}{3}$ $T_3 - T_2 = \frac{47}{12} - \frac{31}{12} = \frac{16}{12} = 1\frac{1}{3}$ $T_4 - T_3 = \frac{63}{12} - \frac{47}{12} = \frac{16}{12} = 1\frac{1}{3}$
• Step 3 (Project the next term): $T_5 = \frac{63}{12} + \frac{16}{12} = \frac{79}{12} = 6\frac{7}{12}$

Notice how the unsimplified fractions reveal a clear, constant numerator addition of $+16$.

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Geometric Progressions (Exponential Ratio Series)

A geometric progression is a sequence where each term after the first is derived by multiplying the preceding term by a fixed, non-zero constant known as the common ratio ($r$).

$\text{General Term: } T_n = a \cdot r^{n-1}$ $\text{Common Ratio: } r = \frac{T_n}{T_{n-1}}$

Common Ratio Typologies

• Whole-Number Multipliers ($r > 1$): Sequences that accelerate violently (e.g., $\times 3$: $2, 6, 18, 54, 162, 486$). When terms double or triple at each step, linear subtraction is a waste of time.
• Unit Fraction Multipliers ($0 < r < 1$): Sequences that contract rapidly toward zero (e.g., $\times \frac{1}{2}$ or $\div 2$: $192, 96, 48, 24, 12, 6$).
• Non-Unit Fractional Multipliers: High-frequency selective entry items use multipliers like $1.5$ ($\times \frac{3}{2}$), $2.5$ ($\times \frac{5}{2}$), or $\frac{2}{3}$.
• Alternating Signs ($r < 0$): When the common ratio is negative, terms alternate between positive and negative values (e.g., $\times (-2)$: $5, -10, 20, -40, 80, -160$).

Worked Example: Non-Unit Fractional Multiplier

$\text{Find the missing term: } 32,\ 48,\ 72,\ 108,\ 162,\ ?$

• Step 1 (Check first differences): $48 - 32 = 16$; $72 - 48 = 24$; $108 - 72 = 36$; $162 - 108 = 54$. The differences are increasing ($16, 24, 36, 54$), showing the pattern is not arithmetic.
• Step 2 (Test common ratios): $\frac{48}{32} = \frac{3}{2} = 1.5$ $\frac{72}{48} = \frac{3}{2} = 1.5$ $\frac{108}{72} = \frac{3}{2} = 1.5$
• Step 3 (Compute the target term): $T_6 = 162 \times \frac{3}{2} = (162 \div 2) \times 3 = 81 \times 3 = 243$

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Two-Tier (Second-Order) Differences ($\Delta_1, \Delta_2$)

When the first differences between consecutive terms ($\Delta_1$) are not constant, do not abandon subtraction. If the first differences themselves form an arithmetic progression, the sequence is a second-order (quadratic) sequence.

In mathematics, a quadratic sequence is defined by the polynomial: $T_n = an^2 + bn + c$

The algebraic behavior across difference tiers is invariant:

• The first difference is linear: $\Delta_1(n) = T_{n+1} - T_n = 2an + (a + b)$
• The second difference is constant: $\Delta_2 = \Delta_1(n+1) - \Delta_1(n) = 2a$

$\text{Key Theorem: } a = \frac{\Delta_2}{2}$

If the second difference $\Delta_2$ is $+6$, then $a = 3$, meaning the underlying sequence involves $3n^2$.

Terms (T):       4          11          22          37          56          79
                  \        /  \        /  \        /  \        /  \        /
First Diff (Δ1):     +7          +11         +15         +19         +23
                       \        /   \       /   \       /   \       /
Second Diff (Δ2):          +4          +4          +4          +4   (Constant!)

Worked Example: Deciphering a Quadratic Pattern

$\text{Find the missing term: } 4,\ 11,\ 22,\ 37,\ 56,\ 79,\ ?$

• Step 1 (First Differences $\Delta_1$): $11 - 4 = +7$ $22 - 11 = +11$ $37 - 22 = +15$ $56 - 37 = +19$ $79 - 56 = +23$
• Step 2 (Second Differences $\Delta_2$): $11 - 7 = +4$ $15 - 11 = +4$ $19 - 15 = +4$ $23 - 19 = +4$
• Step 3 (Extend $\Delta_1$ and calculate target term): $\text{Next } \Delta_1 = 23 + 4 = 27$ $T_7 = 79 + 27 = 106$

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Combined Operation Progressions ($a \cdot T_{n-1} + b$)

Many challenging sequence questions do not conform to pure additions or pure ratios. Instead, each term is generated by applying two sequential arithmetic operations to the preceding term.

$\text{Recursive Form: } T_n = a \cdot T_{n-1} + b_n$

Common operation structures include:

1. Constant multiplier, constant addend: $T_n = 2T_{n-1} + 3$
2. Constant multiplier, increasing addend: $T_n = 2T_{n-1} + n$ (e.g., $\times 2 + 1, \times 2 + 2, \times 2 + 3, \dots$)
3. Constant multiplier, alternating sign addend: $T_n = 2T_{n-1} \pm k$ (e.g., $\times 2 + 1, \times 2 - 1, \times 2 + 1, \dots$)

Worked Example: Combined Operations with Increasing Addend

$\text{Find the missing term: } 3,\ 7,\ 16,\ 35,\ 74,\ ?$

• Analysis of Transitions:
  • From $3$ to $7$: $3 \times 2 = 6 \xrightarrow{+1} 7$
  • From $7$ to $16$: $7 \times 2 = 14 \xrightarrow{+2} 16$
  • From $16$ to $35$: $16 \times 2 = 32 \xrightarrow{+3} 35$
  • From $35$ to $74$: $35 \times 2 = 70 \xrightarrow{+4} 74$
• Identify Rule: Multiply by $2$ and add an incrementing integer ($+1, +2, +3, +4$).
• Calculate Target Term: $T_6 = 74 \times 2 + 5 = 148 + 5 = 153$

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The 15-Second Triage Protocol

When you reach a series item, start testing relationships immediately rather than staring at the numbers. Follow this precise decision tree:

[Step 1: Write first 4 terms with generous spacing (0–3s)]
                       │
                       ▼
[Step 2: Compute first 3 gaps (Δ1) immediately (3–8s)]
       ├── Constant difference (+d or -d)? ──► ARITHMETIC (Done in 10s)
       └── Constant ratio (×r or ÷r)?       ──► GEOMETRIC (Done in 12s)
                       │ (Neither constant?)
                       ▼
[Step 3: Compute second differences (Δ2) (8–14s)]
       └── Constant second difference?       ──► TWO-TIER QUADRATIC (Done in 18s)
                       │ (Δ2 not constant?)
                       ▼
[Step 4: Pivot to Operations or Interleaving (14–20s)]
       ├── Terms approximately doubling/tripling? ──► Test (×a ± b)
       └── Terms oscillating up and down?         ──► Pivot to Interleaved (Section 4.2)

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Progression Types Reference Table

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High-Yield Traps in Numerical Sequences

1. The Two-Term Assumption Trap

A candidate notices that the transition from $2$ to $6$ is $+4$, and prematurely adds $4$ to the next term without verifying the third and fourth terms. In selective entry exams, $2, 6, \dots$ could lead to $18$ ($\times 3$), $12$ ($+4, +6, +8$), or $14$ ($\times 2 + 2$). Always verify your candidate rule across at least three transitions before committing.

2. The Arithmetic Default on Multiplicative Series

When untrained candidates see numbers growing rapidly ($4, 12, 36, 108$), they instinctively write down differences ($+8, +24, +72$) and then waste 20 seconds computing second differences ($+16, +48$). Train your eye to spot multiplicative ratios first whenever terms grow by multiples rather than steady increments.

3. Mismanaging Decreasing Sequences Below Zero

When a negative common difference crosses zero (e.g., $8, 3, -2, -7$), students frequently make sign errors under time stress, writing $-2 - 5 = -3$ (instead of $-7$) or $-7 - 5 = -2$ (instead of $-12$). Remember that subtracting a positive value from a negative number moves further to the left on the number line.