7.3 Missing and Wrong Number Series

7.3 Missing and Wrong Number Series

Number Series questions are a perennial fixture in the State Bank of India (SBI) Junior Associate examination, typically contributing a set of about 5 questions in Prelims (14.3% of the 35-question Numerical Ability section) and appearing in Mains as well. The series presented in banking examinations test pattern recognition, numerical intuition, and systematic structural deduction.

Candidates encounter two distinct question archetypes:

1. Missing Number Series: A sequence contains a question mark (?) at an intermediate position or at the terminal end. Candidates must decode the governing algebraic or geometric rule and compute the missing term.
2. Wrong Number Series: A complete sequence of numbers is provided, but exactly one number violates the governing law. Candidates must isolate the single anomalous term. Wrong number series is cognitively more demanding because an incorrect term corrupts the differences on both sides of itself.

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The Growth Slope Diagnostic Framework

Before calculating random differences or trial-and-error operations, analyze the macro-slope of the sequence. The rate of change between the first term and the last term reveals the mathematical family of the series.

Pattern Identification by Slope Analysis

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Arithmetic Patterns & The Two-Tier Difference Method

When the slope is moderate, the series is governed by arithmetic addition or subtraction. Many SBI Clerk problems do not use a constant first-tier difference; instead, the pattern emerges at the second tier (Double Difference).

The Mathematical Mechanism of Double Difference

Let the sequence be $T_1, T_2, T_3, T_4, T_5, \dots$

• First-tier differences: $\Delta_{1, k} = T_{k+1} - T_k$
• Second-tier differences: $\Delta_{2, k} = \Delta_{1, k+1} - \Delta_{1, k}$

If the second-tier difference is constant ($\Delta_{2, k} = d$), the original series is quadratic: $T_n = an^2 + bn + c$.

Common Second-Tier Progressions in Banking Exams

1. Constant Difference: $\Delta_2 = +3, +3, +3, \dots$
2. Arithmetic Progression: $\Delta_2 = +4, +8, +12, +16, \dots$
3. Consecutive Primes: $\Delta_1 = 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31$ (Beware: 9, 15, 21, 25, 27 are NOT primes!).
4. Triangular Numbers: Differences follow $T_k = \frac{k(k+1)}{2} \implies +1, +3, +6, +10, +15, +21, \dots$

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Geometric, Decimal & Mixed Operations

In clerical exams, pure geometric progressions ($2, 6, 18, 54$) are rare because they are too easily spotted. Setters heavily favor Mixed Geometric-Arithmetic Operations.

1. The Decimal Fractional Step Pattern

Whenever a series begins with an even number, dips slightly or stays nearly equal, and then accelerates into decimals, it almost universally adheres to the Half-Step Multiplier Rule: $T_{k+1} = T_k \times m_k + c_k$ where $m_k \in {0.5, 1.0, 1.5, 2.0, 2.5, 3.0}$.

Classic Structural Trace:

• $24 \times 0.5 + 1 = 13$
• $13 \times 1.0 + 1 = 14$
• $14 \times 1.5 + 1 = 22$
• $22 \times 2.0 + 1 = 45$
• $45 \times 2.5 + 1 = 113.5$

2. The $(N \times k \pm k)$ and $(N \times k \pm m)$ Patterns

• Pattern A (Constant additive offset): $T_{k+1} = T_k \times 2 + 3$
• Pattern B (Progressive multiplier and offset): $T_{k+1} = T_k \times k + k$
  • $5 \times 1 + 1 = 6$
  • $6 \times 2 + 2 = 14$
  • $14 \times 3 + 3 = 45$
  • $45 \times 4 + 4 = 184$
• Pattern C (Alternating sign offset): $T_{k+1} = T_k \times 2 - 1, ; T_{k+2} = T_{k+1} \times 2 + 2, \dots$

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Power-Based & Polynomial Series

Series governed by squares and cubes can be applied directly to terms or to the intervening differences:

1. Difference as Squares or Cubes

• Consecutive Squares: $\Delta_1 = 1^2, 2^2, 3^2, 4^2, 5^2 \implies +1, +4, +9, +16, +25$
• Consecutive Cubes: $\Delta_1 = 1^3, 2^3, 3^3, 4^3, 5^3 \implies +1, +8, +27, +64, +125$
• Alternating Power Adjustments: $\Delta_1 = (1^3 + 1), (2^3 - 1), (3^3 + 1), (4^3 - 1) \implies +2, +7, +28, +63$

2. Direct Power Formulations ($n^3 \pm n$ or $n^2 \pm n$)

Familiarity with the signature numbers of $n^3 \pm n$ allows immediate recognition:

• $n^3 - n$ Signature Numbers: $0, 6, 24, 60, 120, 210, 336, 504, 720$
• $n^3 + n$ Signature Numbers: $2, 10, 30, 68, 130, 222, 350, 520, 738$
• $n^2 + n$ (Pronic Numbers): $2, 6, 12, 20, 30, 42, 56, 72, 90, 110$

If you see the numbers $6, 24, 60, 120, 210$ in a series, do not calculate differences—recognize instantly that $T_n = n^3 - n$.

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The Wrong Number Decoupling Protocol

Finding the wrong number in a series is notoriously challenging because an aspirant checking differences sees two anomalous values and often picks the wrong candidate.

The Twin Disruption Rule

Suppose a sequence has terms $T_1, T_2, T_3, T_4, T_5, T_6$. If term $T_4$ is incorrect:

• $\Delta_1 = T_2 - T_1$ is Normal / Correct
• $\Delta_2 = T_3 - T_2$ is Normal / Correct
• $\Delta_3 = T_4 - T_3$ is CORRUPTED (involves faulty $T_4$)
• $\Delta_4 = T_5 - T_4$ is CORRUPTED (involves faulty $T_4$)
• $\Delta_5 = T_6 - T_5$ is Normal / Correct

[!WARNING] The Common Vertex Invariant: When evaluating first-tier differences in a Wrong Number Series with a wrong interior term, exactly two consecutive differences will fail the expected pattern, and the wrong number is the shared vertex between them (if only the first or the last difference fails, suspect the first or last term): $\text{Wrong Number} = \Delta_{\text{faulty } 1} \cap \Delta_{\text{faulty } 2}$ Never select the term preceding the first faulty difference or following the second faulty difference.

Systematic 4-Step Wrong Number Resolution Protocol

1. Step 1: Compute All Adjacent Differences: Write out the complete list of first-tier differences $\Delta_1, \Delta_2, \Delta_3, \dots$
2. Step 2: Detect the Faulty Pair: Identify the boundary where the mathematical logic breaks down. Locate the two adjacent difference values that violate the sequence rule.
3. Step 3: Isolate the Common Term: The term situated precisely between the two corrupted differences is the incorrect number.
4. Step 4: Verify the Fix: Replace the wrong number with the mathematically correct value. Verify that both corrupted differences immediately transform into the correct expected values.

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Worked Examples: Missing and Wrong Number Series

Example 1: Missing Number with Double Difference

Problem: Find the missing value (?) in the series: $12, \; 19, \; 35, \; 64, \; 110, \; ?$

Solution Walkthrough:

1. Calculate Tier-1 differences ($\Delta_1$):
   • $19 - 12 = 7$
   • $35 - 19 = 16$
   • $64 - 35 = 29$
   • $110 - 64 = 46$
   • First-tier differences: $7, 16, 29, 46$
2. Calculate Tier-2 differences ($\Delta_2$):
   • $16 - 7 = 9$
   • $29 - 16 = 13$
   • $46 - 29 = 17$
   • Second-tier differences: $9, 13, 17$
3. Deduce pattern: Tier-2 forms an arithmetic progression with a common difference of $+4$.
4. Compute next Tier-2 difference: $17 + 4 = 21$.
5. Compute next Tier-1 difference: $46 + 21 = 67$.
6. Compute missing term (?): $110 + 67 = \mathbf{177}$.

Example 2: Missing Number with Decimal Progression

Problem: Find the missing value (?) in the series: $13, \; 7, \; 8, \; 13.5, \; 29, \; ?$

Solution Walkthrough:

1. Notice the initial dip: $13 \rightarrow 7$. Test half-step multipliers:
   • $13 \times 0.5 + 0.5 = 6.5 + 0.5 = 7$
   • $7 \times 1.0 + 1 = 7 + 1 = 8$
   • $8 \times 1.5 + 1.5 = 12 + 1.5 = 13.5$
   • $13.5 \times 2.0 + 2.0 = 27 + 2 = 29$
2. The multiplier increases by $0.5$ at each step, and the additive constant equals the multiplier ($T_n = T_{n-1} \times k + k$):
   • Next factor $k = 2.5$.
   • Next term: $29 \times 2.5 + 2.5 = 72.5 + 2.5 = \mathbf{75.0}$.

Example 3: Wrong Number Series (Twin Disruption Analysis)

Problem: Identify the incorrect number in the series: $6, \; 15, \; 42, \; 120, \; 366, \; 1095$

Solution Walkthrough:

1. Test geometric relationships between adjacent terms:
   • $6 \times 3 - 3 = 18 - 3 = 15$
   • $15 \times 3 - 3 = 45 - 3 = 42$
   • $42 \times 3 - 3 = 126 - 3 = 123$ (The given term is 120!)
   • If the term is $123$: $123 \times 3 - 3 = 369 - 3 = 366$ (Matches!)
   • $366 \times 3 - 3 = 1,098 - 3 = 1,095$ (Matches!)
2. Verify via the Twin Disruption Rule on differences:
   • $15 - 6 = 9 = 3^2$
   • $42 - 15 = 27 = 3^3$
   • $120 - 42 = 78$ (Corrupted! Should be $3^4 = 81$)
   • $366 - 120 = 246$ (Corrupted! Should be $3^5 = 243$)
   • $1,095 - 366 = 729 = 3^6$
3. The two corrupted differences are 78 and 246. The common vertex sharing these two differences is 120.
4. Therefore, the wrong number is 120 (it should be replaced by 123).