9.2 Number & Letter Series, Coding-Decoding & Sequence Completion
Key Takeaways
- Number and letter series test pattern recognition across arithmetic increments, geometric ratios, prime sequences, and alternating dual-track series.
- Memorizing English letter positions (A=1 to Z=26) and their reverse pairs (A-Z, B-Y, C-X) allows instant calculation without drawing alphabet charts during the test.
- Coding-decoding questions rely on fixed numerical shifts (+k, -k), opposite letter substitutions, or positional matrix rearrangements.
- When solving multi-step series, calculating the first-order difference (Δ1) and second-order difference (Δ2) reveals hidden polynomial patterns.
- In speed tests, checking the first and last letters of a coded string eliminates incorrect distractor options in under 10 seconds.
Last updated: July 2026
9.2 Number & Letter Series, Coding-Decoding & Sequence Completion
Number series, letter sequences, and coding-decoding problems form the core of numerical and symbolic logic in the Pakistan Army Medical Cadet (AMC) Initial Intelligence Test. These questions evaluate a candidate's mental agility, numerical pattern recognition, and speed under strict time limits. With less than 21 seconds allocated per question, mastering systematic solving algorithms is essential.
1. Classification of Number Series Patterns
Number series consist of numbers ordered according to a specific logical rule. To find the missing term, candidates must systematically test the primary series types:
A. Arithmetic (Difference-Based) Series
In arithmetic series, successive terms increase or decrease by a constant or uniformly changing value ($+d$ or $-d$).
- Constant Difference: $4, 9, 14, 19, 24, \dots (+5)$
- Increasing Difference: $2, 5, 10, 17, 26, \dots (+3, +5, +7, +9)$
- Decreasing Difference: $50, 48, 44, 38, 30, \dots (-2, -4, -6, -8)$
B. Geometric (Ratio-Based) Series
Terms are multiplied or divided by a constant factor ($\times k$ or $\div k$).
- Example: $3, 6, 12, 24, 48, \dots (\times 2)$
- Alternating Multipliers: $2, 6, 18, 54, 162, \dots (\times 3)$
C. Polynomial & Difference of Differences ($\Delta_2$) Series
When first-order differences ($\Delta_1$) do not yield an obvious pattern, calculate second-order differences ($\Delta_2$):
\text{Series:} & & 2 & & 6 & & 12 & & 20 & & 30 & & \mathbf{42} \\ \Delta_1: & & & +4 & & +6 & & +8 & & +10 & & \mathbf{+12} & \\ \Delta_2: & & & & +2 & & +2 & & +2 & & \mathbf{+2} & & \end{array}$$ ### D. Square, Cube & Prime Series Series built on exponential powers ($n^2 \pm k$ or $n^3 \pm k$) or prime numbers: * **Squares**: $1, 4, 9, 16, 25, 36, \dots (n^2)$ * **Square $\pm 1$**: $0, 3, 8, 15, 24, 35, \dots (n^2 - 1)$ * **Primes**: $2, 3, 5, 7, 11, 13, 17, 19, 23, \dots$ ### E. Alternating / Interleaved Dual Series Two independent series merged into a single sequence at alternating positions (Odd terms = Series 1; Even terms = Series 2). * **Example**: $5, 20, 10, 18, 15, 16, \mathbf{20}, \mathbf{14}$ * *Track A (Odd positions)*: $5, 10, 15, \mathbf{20} \quad (+5)$ * *Track B (Even positions)*: $20, 18, 16, \mathbf{14} \quad (-2)$ --- ## 2. Letter Series & Positional Alphabet Values Letter series questions replace numbers with letters from the English alphabet. Instant recall of numerical letter positions eliminates the need to write down the alphabet during the exam. ### The Standard Positional Value Chart Memory tools like **EJOTY** give quick reference benchmarks for every 5th letter: | Letter | **E** | **J** | **O** | **T** | **Y** | | :---: | :---: | :---: | :---: | :---: | :---: | | **Position** | 5 | 10 | 15 | 20 | 25 | #### Complete Alphabet Numerical Values: $$\begin{array}{ccccccccccccc} \text{A=1} & \text{B=2} & \text{C=3} & \text{D=4} & \text{E=5} & \text{F=6} & \text{G=7} & \text{H=8} & \text{I=9} & \text{J=10} & \text{K=11} & \text{L=12} & \text{M=13} \\ \text{N=14} & \text{O=15} & \text{P=16} & \text{Q=17} & \text{R=18} & \text{S=19} & \text{T=20} & \text{U=21} & \text{V=22} & \text{W=23} & \text{X=24} & \text{Y=25} & \text{Z=26} \end{array}$$ ### Reverse Alphabet Pairs (Sum of 27 Rule) Opposite letters in the alphabet always sum to **27**: $$\text{Position of Letter} + \text{Position of Opposite Letter} = 27$$ * **A (1) $\leftrightarrow$ Z (26)** $\implies 1 + 26 = 27$ * **B (2) $\leftrightarrow$ Y (25)** $\implies 2 + 25 = 27$ * **C (3) $\leftrightarrow$ X (24)** $\implies 3 + 24 = 27$ * **M (13) $\leftrightarrow$ N (14)** $\implies 13 + 14 = 27$ ### Letter Skip Patterns Letter sequences often skip a fixed or growing number of intermediate letters: * **Example**: `B, E, H, K, N, ?`-------- * Convert to numbers: $2, 5, 8, 11, 14 \implies \text{Pattern is } +3$. * Next number: $14 + 3 = 17 \implies \mathbf{Q}$. --- ## 3. Coding-Decoding Mechanics Coding-decoding tests convert plain text into ciphertext based on systematic rules. The AMC test utilizes three main types of coding: ### A. Letter Shifting / Substitution Coding Each letter in the word is shifted forward ($+k$) or backward ($-k$) by a fixed number of positions in the alphabet. * **Example**: If `CADET` is coded as `DBEFU`: * $C (+1) \rightarrow D$ * $A (+1) \rightarrow B$ * $D (+1) \rightarrow E$ * $E (+1) \rightarrow F$ * $T (+1) \rightarrow U$ * **Applying to `ARMY`**: * $A (+1) \rightarrow B$, $R (+1) \rightarrow S$, $M (+1) \rightarrow N$, $Y (+1) \rightarrow Z \implies \mathbf{BSNZ}$. ### B. Opposite Letter Coding Letters are replaced by their 27-sum inverse counterpart. * **Example**: `BLOOD` $\rightarrow$ `YLOOW`-------- * $B \leftrightarrow Y$, $L \leftrightarrow O$, $O \leftrightarrow L$, $O \leftrightarrow L$, $D \leftrightarrow W$. ### C. Direct Numerical / Matrix Coding Each letter is assigned a specific digit based on position or given reference keys. * **Example**: If `MEDICAL` = `1234567` and `ARMY` = `6819`, then `CARD` is extracted by picking matching digits. --- ## 4. Step-by-Step Worked AMC Exam Examples ### Worked Example 1: Alphanumeric Dual Sequence **Question**: Find the next term in the series: `Z26, X24, V22, T20, ?` * **Step 1 (Analyze Letters)**: Z (26), X (24), V (22), T (20). The letters move backward by 2 steps ($-2$). * Next letter: $20 - 2 = 18 \implies \mathbf{R}$. * **Step 2 (Analyze Numbers)**: $26, 24, 22, 20$. The numbers decrease by 2 ($-2$). * Next number: $20 - 2 = \mathbf{18}$. * **Step 3 (Combine)**: The next term is **R18**. ### Worked Example 2: Complex Number Series **Question**: What number comes next? `3, 7, 15, 31, 63, ?` * **Step 1 (Check Differences)**: $+4, +8, +16, +32$. The difference doubles each step ($\times 2$). * Next difference: $32 \times 2 = 64$. * Next term: $63 + 64 = \mathbf{127}$. * **Alternative Formula**: $T_n = 2 T_{n-1} + 1$. * $2(63) + 1 = 126 + 1 = \mathbf{127}$. --- ## 5. Speed Tactics for AMC Intelligence Test * **First/Last Character Test**: In coding-decoding questions, code only the first and last letters of the word. Usually, this instantly narrows down the 4 options to the single correct answer. * **Spot Dual Series Fast**: If a number series fluctuates up and down (e.g., $10, 5, 12, 4, 14, 3$), it is almost certainly a dual-track series. Split odd and even terms immediately. * **Avoid Writing Full Tables**: Memorize $E=5, J=10, O=15, T=20, Y=25$ so you can compute relative positions mentally in 2 seconds.Test Your Knowledge
What is the next number in the given series? 2, 6, 12, 20, 30, ?
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Test Your Knowledge
Find the missing letter in the sequence: B, E, H, K, N, ?
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Test Your Knowledge
If the word "CADET" is coded as "DBEFU" in a secret code, how is "ARMY" written in that same code?
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Test Your Knowledge
Determine the next term in the alphanumeric sequence: Z26, X24, V22, T20, ?
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