2.2 Number Series (Missing & Wrong Number Patterns)

2.2 Number Series (Missing & Wrong Number Patterns)

Number Series questions carry a consistent weightage of 5 questions in SBI PO Prelims and frequently appear in innovative formats in SBI PO Mains. The candidate is presented with a sequence of numbers adhering to a strict mathematical rule. The objective is to either identify a Missing Number denoted by a question mark (?) or pinpoint a Wrong Number that violates the established pattern.

While missing number series test pattern recognition, wrong number series introduce significantly higher cognitive load because a single wrong entry alters two consecutive difference terms, obscuring the underlying logic.

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Structural Pattern Classification Framework

Before calculating arithmetic differences, inspect the overall rate of progression across the series. The macro-behavior of the numbers reveals the mathematical family of the pattern.

[Series Growth Rate Inspection]
  ├── Gradual / Slow Increase ────────► Addition or Single Difference
  ├── Fluctuating Growth Rate ────────► Double Difference / Prime Differences
  ├── Rapid Exponential Growth ──────► Multiplication or Geometric Powers (x2, x3...)
  ├── Initial Decrease then Increase ─► Decimal Multiplier (x0.5 + 0.5)
  └── Zig-zag Up and Down ────────────► Alternating / Twin Series

Summary Table of High-Frequency SBI PO Patterns

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Deep-Dive: Missing Number Series Mechanics

When tackling a missing number series, execute the four-step diagnostic procedure:

1. Compute the first-level differences (Primary Difference $\Delta_1$).
2. If $\Delta_1$ exhibits no obvious pattern, compute second-level differences (Secondary Difference $\Delta_2$).
3. Check if $\Delta_1$ terms are perfect squares ($1, 4, 9, 16...$), cubes ($1, 8, 27...$), or prime numbers ($2, 3, 5, 7, 11...$).
4. If terms drop initially (e.g., $16 \rightarrow 9 \rightarrow 10$), test decimal multipliers starting with $\times 0.5$.

Worked Example 1: Increasing Multiplier-Cum-Addition Series

Problem: Find the missing term ? in the sequence: $12, 13, 28, 87, 352, ?$

Step-by-step Solution:

• Inspect growth: $12 \rightarrow 13$ (slow), but $87 \rightarrow 352$ (rapid multiplication).
• Evaluate ratio of adjacent terms:
  • $13 / 12 \approx 1 \implies 12 \times 1 + 1 = 13$
  • $28 / 13 \approx 2 \implies 13 \times 2 + 2 = 28$
  • $87 / 28 \approx 3 \implies 28 \times 3 + 3 = 87$
  • $352 / 87 \approx 4 \implies 87 \times 4 + 4 = 352$
• Apply identified pattern ($T_{n+1} = T_n \times 5 + 5$): $\text{Missing Term} = 352 \times 5 + 5 = 1760 + 5 = \mathbf{1765}$

Worked Example 2: Cube-Minus-Square Pattern Series

Problem: Find the missing term ? in the sequence: $0, 4, 18, 48, 100, 180, ?$

Step-by-step Solution:

• Compute Primary Differences ($\Delta_1$):
  • $4 - 0 = 4$
  • $18 - 4 = 14$
  • $48 - 18 = 30$
  • $100 - 48 = 52$
  • $180 - 100 = 80$
• Compute Secondary Differences ($\Delta_2$):
  • $14 - 4 = 10$
  • $30 - 14 = 16$
  • $52 - 30 = 22$
  • $80 - 52 = 28$
• Observe $\Delta_2$: Arithmetic progression with a constant difference of $+6$.
• Next $\Delta_2 = 28 + 6 = 34$.
• Next $\Delta_1 = 80 + 34 = 114$.
• Next Term $= 180 + 114 = \mathbf{294}$.

Alternative Structural Logic: Express terms directly as $n^3 - n^2$:

• $1^3 - 1^2 = 1 - 1 = 0$
• $2^3 - 2^2 = 8 - 4 = 4$
• $3^3 - 3^2 = 27 - 9 = 18$
• $4^3 - 4^2 = 64 - 16 = 48$
• $5^3 - 5^2 = 125 - 25 = 100$
• $6^3 - 6^2 = 216 - 36 = 180$
• $7^3 - 7^2 = 343 - 49 = \mathbf{294}$.

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Deep-Dive: Wrong Number Series Mechanics

In Wrong Number Series, the sequence contains one invalid term that violates the underlying rule. Recognizing how errors propagate is the key to solving these questions under 30 seconds.

The Error Propagation Property

Suppose a series adheres to the rule $T_{n+1} = T_n + d_n$. If term $T_k$ is incorrect:

• Difference $D_{k-1} = T_k - T_{k-1}$ will be incorrect.
• Difference $D_k = T_{k+1} - T_k$ will also be incorrect.
• All other differences remain completely correct.

[!IMPORTANT] The Intersection Rule for Wrong Numbers: Calculate all primary differences $\Delta_1$. Identify the two adjacent difference values that fail to fit the logical pattern. The number situated directly between those two faulty differences is guaranteed to be the wrong number.

Sequence:    T1 ─── T2 ─── T3 ─── [T4-WRONG] ─── T5 ─── T6
Differences:   d1    d2    [d3-FAULTY]  [d4-FAULTY]   d5
                           ▲                      ▲
                           └────── Intersection ──┘
                                 Isolates T4!

Worked Example 3: Wrong Number Isolation

Problem: Identify the wrong number in the series: $6, 14, 31, 64, 130, 262$.

Step-by-step Solution:

• Test multiplication rule $T_{n+1} = T_n \times 2 + 2$:
  • $6 \times 2 + 2 = 14$ (Correct)
  • $14 \times 2 + 2 = \mathbf{30}$ (Series lists 31)
  • Test with correct term 30: $30 \times 2 + 2 = 62$... wait, let's analyze differences:
• Compute primary differences:
  • $14 - 6 = 8$
  • $31 - 14 = 17$ (Faulty)
  • $64 - 31 = 33$ (Faulty)
  • $130 - 64 = 66$
  • $262 - 130 = 132$
• Analyze valid differences: $8 \rightarrow 66 \rightarrow 132$. The sequence of differences doubles at each step ($8, 16, 32, 64, 128$).
• The two faulty differences are 17 and 33.
• The term positioned between 17 and 33 is 31.
• Replacing 31 with 30 gives differences $30 - 14 = 16$ and $62 - 30 = 32$, which restores the perfect doubling difference pattern.
• Therefore, 31 is the wrong number.

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Advanced Mains Pattern: Parallel Derived Series

In SBI PO Mains, questions often present a complete Series A followed by the start of Series B.

Example:

• Series A: $8, 14, 38, 98, 222, 446$
• Series B: $5, (a), (b), (c), (d), (e)$

Task: Find the value of $(c)$ if Series B follows the exact same logic as Series A.

1. Decode Series A logic: Primary differences are $6, 24, 60, 124, 224$. Express as $n^3 - n$ ($2^3-2=6, 3^3-3=24, 4^3-4=60, 5^3-5=124, 6^3-6=216...$).
2. Apply identical difference sequence starting from 5 to calculate $(a), (b),$ and $(c)$ step-by-step.