What Is a Series?
A Series (or Sequence) is an ordered succession of terms—numbers, letters, words, or alphanumeric clusters—generated according to a definite underlying mathematical, positional, or logical relationship. The solver's task is to uncover that generative law and use it to determine a missing term, find the next term, or identify a rogue term that violates the rule.
Systematic Pattern Detection
Series reasoning is not random guessing; it is controlled hypothesis testing. A genuine rule must account consistently for every single term in the sequence, not just the last pair. By systematically evaluating differences, multiplicative ratios, powers, prime properties, and position indices, you isolate the true generative law with 100% mathematical certainty.
In management entrance exams (CAT, MBA CET, SNAP, NMAT, XAT), every series question falls into one of five structural categories:
Single uniform operator applied step-by-step (e.g. constant addition $+7$, geometric ratio $\times 3$).
Two independent sub-series interwoven at odd ($1, 3, 5$) and even ($2, 4, 6$) indices.
Compound formulas combining operations like $T_n = 2 T_{n-1} + n$ or alternating between $+k$ and $\times m$.
Letters mapped to numbers $A=1 \dots Z=26$, reverse positions ($27-\text{rank}$), or modulo 26 cyclic jumps.
Sequences where short prefixes admit multiple rules; resolved strictly by the Principle of Parsimony.
Series questions test rapid pattern recognition, numerical fluency, and mental arithmetic. In speed-based exams like SNAP and MBA CET, cracking 4-6 series questions in under 3 minutes provides a decisive competitive edge.
Heavily featured in MAH MBA CET (5-8 standalone questions), SNAP (3-5 questions), NMAT (Language & Logical reasoning), XAT (Decision Making sequences), and CAT LRDI set pattern foundations.
Familiarity with prime numbers up to 100, perfect squares up to $35^2$, perfect cubes up to $15^3$, basic modular arithmetic, and the alphabetical index mapping ($A=1$ through $Z=26$).
Number Series Fundamentals
Before testing complex formulas, an expert reasoning solver inspects seven elementary relational indicators between terms:
Subtract each term from its successor: $d_i = T_{i+1} - T_i$. If $d_i$ is constant, it is an arithmetic progression.
When first differences increase or decrease at a steady pace, take the difference of differences: $\Delta_2 = d_{i+1} - d_i$.
When terms grow aggressively by multiples, evaluate $r_i = \frac{T_{i+1}}{T_i}$. Constant ratio indicates a geometric series.
Successive steps alternate between two distinct operations (e.g., $+a, \times b, +a, \times b$).
A term depends on the sum, product, or difference of previous terms (e.g., Fibonacci recurrence: $T_n = T_{n-1} + T_{n-2}$).
Terms do not relate to immediate neighbours, but connect strictly to terms at index $n \pm 2$ or $n \pm 3$.
Mathematically, any finite set of $k$ numbers can be fitted by an infinite number of high-order polynomial functions. However, competitive aptitude exams operate under the Principle of Parsimony: the simplest generative rule that consistently explains every provided term without arbitrary ad-hoc exceptions is the intended answer. If both a simple linear difference rule and an intricate 4th-degree polynomial fit the prefix, always select the simpler, foundational rule.
12 Common Number Series Patterns
Constant Addition / Subtraction
How to Recognise: Terms increase or decrease steadily by an identical increment $d$.
How to Test: Compute $T_2 - T_1, T_3 - T_2, \dots$ Confirm that all differences match exactly.
Differences: $+7, +7, +7, +7 \implies \text{Next} = 36 + 7 = \mathbf{43}$.
Constant Multiplication / Division
How to Recognise: Consecutive terms explode in magnitude or shrink rapidly by a fixed factor.
How to Test: Check the ratio $r = \frac{T_{k+1}}{T_k}$. Verify equality across all terms.
Ratios: $\times 3, \times 3, \times 3 \implies \text{Next} = 108 \times 3 = \mathbf{324}$.
Increasing Differences
How to Recognise: Step sizes grow gradually ($+2, +4, +6, +8$).
How to Test: Compute first differences; check if they form an AP, consecutive evens, or primes.
Differences: $+2, +4, +6, +8 \implies \text{Next diff} = +10 \implies 25 + 10 = \mathbf{35}$.
Decreasing Differences
How to Recognise: Successive gaps between terms contract steadily.
How to Test: Subtract terms; observe if differences diminish by a uniform decrement.
Differences: $-15, -13, -11, -9 \implies \text{Next diff} = -7 \implies 52 - 7 = \mathbf{45}$.
Squares & Square Offsets
How to Recognise: Numbers hover close to famous perfect squares ($1, 4, 9, 16, 25, 36, 49 \dots$).
How to Test: Express each term as $n^2, n^2-1, n^2+1, \text{ or } n^2-n$.
Rule: $n^2 - 1$ for $n = 2, 3, 4, 5, 6 \implies \text{Next} = 7^2 - 1 = \mathbf{48}$.
Cubes & Cube Offsets
How to Recognise: Numbers hover near perfect cubes ($1, 8, 27, 64, 125, 216, 343 \dots$).
How to Test: Formulate as $n^3, n^3 \pm 1, n^3 \pm n$, or $n^3 + n^2$.
Rule: $n^3 + 1$ for $n = 1, 2, 3, 4, 5 \implies \text{Next} = 6^3 + 1 = 216 + 1 = \mathbf{217}$.
Prime Number Patterns
How to Recognise: Sequence contains numbers with exactly two distinct positive divisors ($2, 3, 5, 7, 11, 13, 17 \dots$).
How to Test: Verify that no composite numbers (such as $9, 15, 21, 27$) are present.
Consecutive primes $\implies \text{Next prime after } 29 = \mathbf{31}$.
Multiples & Divisibility
How to Recognise: All terms share a distinct common factor or obey modulo constraints.
How to Test: Check whether terms are successive multiples $k \times m$, or share specific divisibility remainders.
Multiples of 14: $14 \times 1, \dots, 14 \times 5 \implies 14 \times 6 = \mathbf{84}$.
Triangular & Figurate Numbers
How to Recognise: Sum of the first $n$ natural numbers: $T_n = \frac{n(n+1)}{2}$.
How to Test: Differences between terms are $2, 3, 4, 5, 6 \dots$ starting from 1.
Gaps: $+2, +3, +4, +5 \implies \text{Next gap } +6 \implies 15 + 6 = \mathbf{21}$.
Factorial Progressions
How to Recognise: Super-exponential growth: $1, 2, 6, 24, 120, 720 \dots$
How to Test: Consecutive multipliers grow by 1 each step: $\times 2, \times 3, \times 4, \times 5$.
Standard: $1, 2, 6, 24, 120, ? \implies \times 2, \times 3, \times 4, \times 5 \implies 120 \times 6 = \mathbf{720}$.
Digit Sums & Products
How to Recognise: Standard arithmetic differences fail, but individual digits reveal internal structure.
How to Test: Sum digits ($S(n)$) or multiply digits ($P(n)$). Check $T_n = T_{n-1} + S(T_{n-1})$.
Rule: $T_n = T_{n-1} + \text{sum of digits}(T_{n-1})$
$12 + (1+2) = 15; 15 + (1+5) = 21; 21 + (2+1) = 24; 24 + (2+4) = 30 \implies 30 + (3+0) = \mathbf{33}$.
Combined Operations
How to Recognise: Terms roughly double, triple, or halve with a constant or stepping additive offset.
How to Test: Test linear recurrence $T_n = m \cdot T_{n-1} \pm c$.
Rule: $\times 2 + 1 \implies 47 \times 2 + 1 = \mathbf{95}$.
Difference-Based Series & Difference Trees
The Difference Tree is the single most powerful diagnostic tool in number series. When a series does not show an immediate constant difference or ratio, calculating successive tiers of differences reveals its polynomial degree.
Consider the sequence: $2, 5, 10, 17, 26, ?$
| Tier | Sequential Terms & Differences | |||||
|---|---|---|---|---|---|---|
| Original Series ($T_n$) | 2 | 5 | 10 | 17 | 26 | ? = 37 |
| 1st Differences ($\Delta_1$) | +3 | +5 | +7 | +9 | +11 | |
| 2nd Differences ($\Delta_2$) | — | +2 | +2 | +2 | +2 (CONSTANT) | |
- If the 1st difference is constant, $T_n$ is linear: $T_n = an + b$.
- If the 2nd difference is constant, $T_n$ is quadratic: $T_n = an^2 + bn + c$ (with $\Delta_2 = 2a$). Here, $\Delta_2 = 2 \implies a = 1$, giving $n^2 + 1$.
- If the 3rd difference is constant, $T_n$ is cubic: $T_n = an^3 + bn^2 + cn + d$ (with $\Delta_3 = 6a$).
Never blindly construct difference trees on geometric, factorial, or alternating sequences! If terms multiply ($\times 2, \times 3$), taking differences merely creates another geometric series without collapsing to a constant. Stop and check ratios if differences grow exponentially.
Alternating & Interleaved Series
When a series oscillates up and down (e.g., $15, 30, 18, 26, 21, 22 \dots$), consecutive differences will fluctuate chaotically between positive and negative numbers. This is the classic signature of an Interleaved Series: two completely separate sequences merged into one.
Analyze: $4, 18, 8, 14, 12, 10, 16, ?$
Missing-Term Series (Single & Multiple Blanks)
In a missing-term problem, one or two values in the middle or at the end of the sequence are replaced with a question mark ($?$). To avoid false matches, follow this 5-step checklist:
Inspect terms right before & after the blank.
Assess if growth is additive ($\Delta_1$) or multiplicative ($r$).
Check against squares, cubes, primes, and digit sums.
Check whether odd and even positions obey different rules.
Crucial: ensure proposed term produces the term after the blank.
Test linear recurrence: $T_n = 2 T_{n-1} + 2$.
$3 \times 2 + 2 = 8$; $8 \times 2 + 2 = 18$; $18 \times 2 + 2 = 38$.
Calculate blank: $38 \times 2 + 2 = \mathbf{78}$.
Bridging Verification: $78 \times 2 + 2 = 156 + 2 = 158$. Exactly matches the post-blank term!
Notice $23 - 17 = 6$ and $35 - 5 = 30$ over 5 steps ($30/5 = 6$).
Hypothesis: Constant AP with $d = +6$.
First gap: $5 + 6 = \mathbf{11}$. (Check: $11 + 6 = 17$, valid).
Second gap: $23 + 6 = \mathbf{29}$. (Check: $29 + 6 = 35$, valid).
Both blanks confirmed: $(11, 29)$.
Incorrect-Term Series ("Find the Wrong Number")
"Find the Wrong Number" is notoriously harder than missing-number questions because test-takers do not know where the rule breaks. However, there is an exact mathematical theorem that solves these instantly:
In any difference-based sequence, an error in a single term $T_k$ corrupts exactly two adjacent differences: $d_{k-1} = T_k - T_{k-1}$ and $d_k = T_{k+1} - T_k$. The term at the intersection of the two anomalous differences is always the faulty number!
Alphabet Series & Letter Jumps
Alphabet series are simply numerical series disguised as letters. By memorizing positional anchors, you translate letter sequences into standard integers in 2 seconds flat.
5
10
15
20
25
Use EJOTY to pinpoint neighbouring letters immediately (e.g., $S$ is 1 before $T \implies 19$; $L$ is 2 after $J \implies 12$).
$Z = 27 - 26 = 1$; $A = 27 - 1 = 26$; $M = 27 - 13 = 14$. Opposite pairs always sum to 27 ($A-Z, B-Y, C-X, D-W, E-V, \dots$).
When forward increments exceed 26, wrap cyclically using modulo 26 arithmetic:
The next difference is $+7$. Starting from $T(20)$: $20 + 7 = 27$.
Applying modulo 26: $27 - 26 = 1 \implies \mathbf{A}$. The next letter is $\mathbf{A}$.
Alphanumeric Series
Alphanumeric series combine letters and digits into compound clusters (e.g. $A2Z, C5X, F10U, J17Q, ?$). The golden rule: decouple the components. Treat each character slot as an independent parallel progression before investigating any internal cross-coupling.
Next increment: $+5$
$10 + 5 = 15 \implies \mathbf{O}$
Next increment: $+9$ (or $n^2+1$)
$17 + 9 = \mathbf{26}$
Next decrement: $-5$
$17 - 5 = 12 \implies \mathbf{L}$
Word & Symbol Series
Unlike Coding & Decoding, which translates words into ciphers, Word & Symbol Series focus strictly on sequential transformations of words based on structural attributes:
Words ordered by character count or vowel counts:
Lengths: 2, 3, 4, 5, 6 $\implies$ Answer must be a 7-letter word (e.g. Freedom).
Initial letters of ordered real-world sequences (days, months, planets):
Months: Jan, Feb, Mar, Apr, May, Jun $\implies$ Next is J (July).
The terminal letter of each word becomes the initial letter of the next:
Ends with D $\implies$ Next word must start with D (e.g. DEER).
Advanced & Hybrid Series
High-percentile CAT and XAT questions employ Hybrid Series that integrate multiple operators, position index dependencies, and Fibonacci-style multi-term recurrences.
Each term is the sum of the preceding 3 terms ($T_n = T_{n-1} + T_{n-2} + T_{n-3}$):
Verification:
$1+1+2 = 4$
$1+2+4 = 7$
$2+4+7 = 13$
$4+7+13 = 24$
Next: $7 + 13 + 24 = \mathbf{44}$.
The multiplier or additive constant increments with the term index $n$:
Step 1 ($n=1$): $4 \times 2 + 1 = 9$
Step 2 ($n=2$): $9 \times 2 + 2 = 20$
Step 3 ($n=3$): $20 \times 2 + 3 = 43$
Step 4 ($n=4$): $43 \times 2 + 4 = 90$
Step 5 ($n=5$): $90 \times 2 + 5 = 180 + 5 = \mathbf{185}$.
The 11-Step Repeatable Solving Framework
Follow this battle-tested 11-step execution workflow on every series question under timed conditions:
Record the Sequence Accurately
Write down every term cleanly with sufficient whitespace above and below for difference tiers.
Calculate First Differences ($\Delta_1$)
Compute $T_2 - T_1, T_3 - T_2, \dots$ If differences are constant, you have an AP. If they increase linearly, prepare for $\Delta_2$.
Check Ratios and Multiplicative Factors
If numbers jump by factors ($\times 2, \times 3, \times 4$), divide successive terms. Do not waste time computing differences on exponential growth.
Inspect Squares, Cubes, and Primes
Test if terms equal $n^2 \pm k$, $n^3 \pm k$, $n^3 \pm n$, or prime numbers. This resolves 40% of non-linear questions instantly.
Evaluate Second Differences ($\Delta_2$)
If first differences increase steadily, calculate $\Delta_2$. A constant second difference proves a quadratic formula ($an^2+bn+c$).
Split Alternating Tracks (Odd vs Even)
If values oscillate up and down, isolate odd positions ($1, 3, 5$) and even positions ($2, 4, 6$) into independent series.
Convert Letters to Positional Integers
For alphabet series, immediately write ranks $A=1 \dots Z=26$ using EJOTY. Treat cyclic wraps ($Z \to A$) modulo 26.
Decouple Alphanumeric Components
Separate letter sequences from numeric sequences. Solve both parallel tracks independently before checking cross-links.
Verify Candidate Rule Across Every Term
Never finalize an answer that fits only 2 terms. The hypothesized rule must satisfy every provided number in the sequence.
Execute Bridging Check
For missing-term or incorrect-term series, calculate the proposed term and confirm it mathematically reproduces the succeeding term.
Apply Principle of Parsimony
If two rules fit, choose the simpler, more natural rule. If ambiguous under strict time pressure, eliminate impossible options and move forward.
10 Fully Worked Solved Examples
Question: Find the next term in the sequence: $14, 23, 32, 41, 50, ?$
$23 - 14 = +9$
$32 - 23 = +9$
$41 - 32 = +9$
$50 - 41 = +9$
Question: Find the next term in the sequence: $6, 18, 54, 162, ?$
$18 / 6 = 3$
$54 / 18 = 3$
$162 / 54 = 3$
Question: Find the next term in the sequence: $7, 11, 19, 31, 47, ?$
$\Delta_1$: $11 - 7 = 4$; $19 - 11 = 8$; $31 - 19 = 12$; $47 - 31 = 16$.
First differences are: $4, 8, 12, 16$ (Multiples of 4, or $\Delta_2 = +4$).
Question: What is the next term in the series: $2, 9, 28, 65, 126, ?$
$1^3 + 1 = 1 + 1 = 2$
$2^3 + 1 = 8 + 1 = 9$
$3^3 + 1 = 27 + 1 = 28$
$4^3 + 1 = 64 + 1 = 65$
$5^3 + 1 = 125 + 1 = 126$
Question: Find the next term in the sequence: $11, 13, 17, 19, 23, 29, 31, ?$
Prime list: $2, 3, 5, 7, \mathbf{11, 13, 17, 19, 23, 29, 31}, \mathbf{37}, 41 \dots$
Question: Determine the next number in the sequence: $4, 18, 8, 14, 12, 10, 16, ?$
Odd positions (1st, 3rd, 5th, 7th): $4, 8, 12, 16$ (Rule: $+4$ each step).
Even positions (2nd, 4th, 6th, 8th): $18, 14, 10, ?$ (Rule: $-4$ each step).
Question: Find the missing term in the sequence: $3, 8, 18, 38, ?, 158$
Step 1: $3 \times 2 + 2 = 8$
Step 2: $8 \times 2 + 2 = 18$
Step 3: $18 \times 2 + 2 = 38$
(Alternatively, differences are $5, 10, 20, 40, 80$, which double each time).
Question: One term in the given sequence is incorrect. Identify it: $5, 12, 26, 54, 112, 222$
Term 1 to 2: $5 \times 2 + 2 = 12$
Term 2 to 3: $12 \times 2 + 2 = 26$
Term 3 to 4: $26 \times 2 + 2 = 54$
Term 4 to 5: $54 \times 2 + 2 = \mathbf{110}$ (Given term is 112!)
Term 5 to 6: $110 \times 2 + 2 = 222$ (Matches the given 222!)
Question: Find the next term in the letter series: $B, E, I, N, T, ?$
$B = 2$
$E = 5$ (Gap: $5 - 2 = +3$)
$I = 9$ (Gap: $9 - 5 = +4$)
$N = 14$ (Gap: $14 - 9 = +5$)
$T = 20$ (Gap: $20 - 14 = +6$)
Question: Find the next term in the alphanumeric sequence: $A2Z, C5X, F10U, J17Q, ?$
• First letter: $A(1) \xrightarrow{+2} C(3) \xrightarrow{+3} F(6) \xrightarrow{+4} J(10) \xrightarrow{+5} \mathbf{O(15)}$
• Middle number: $2 \xrightarrow{+3} 5 \xrightarrow{+5} 10 \xrightarrow{+7} 17 \xrightarrow{+9} \mathbf{26}$ (also $n^2+1$ for $n=1..5$)
• Third letter: $Z(26) \xrightarrow{-2} X(24) \xrightarrow{-3} U(21) \xrightarrow{-4} Q(17) \xrightarrow{-5} \mathbf{L(12)}$
10 Common Traps & How to Avoid Them
1. Checking Only Consecutive Differences
Trap: Computing $T_2 - T_1$ and assuming an AP without checking ratios or powers.
Fix: If numbers double or multiply, switch immediately to testing ratios.
2. Ignoring Ratios on Rapid Growth
Trap: Calculating multi-tier difference trees on series like $3, 12, 48, 192$.
Fix: When terms grow by factors $> 2$, test geometric division first.
3. Missing an Alternating Pattern
Trap: Trying to fit a single monotonic rule onto a sequence that oscillates up and down.
Fix: Separate terms at odd positions from terms at even positions.
4. Confusing Differences with Ratios
Trap: Confusing $+8, +16, +32$ with $\times 2, \times 2, \times 2$.
Fix: Note that $T_n = 2 T_{n-1}$ produces doubling differences, but the starting base dictates the formula.
5. Assuming Every Curve Is Squares or Cubes
Trap: Forcing $n^2$ onto numbers that actually follow prime gaps or Fibonacci sums.
Fix: Memorize prime numbers up to 100 to spot prime series instantly.
6. Forcing an Overly Complex Polynomial
Trap: Inventing a 4-variable cubic equation when a simple digit sum explains the pattern.
Fix: Follow Occam's Razor: test elementary patterns before exotic polynomials.
7. Checking Only the Final Two Terms
Trap: Finding a rule that links $T_{k-1}$ and $T_k$, but failing to verify on $T_1$ and $T_2$.
Fix: Every candidate rule must be validated across the entire given sequence.
8. Applying Cyclic Wraparound Without Proof
Trap: Wrapping $Z \to A$ arbitrarily when the sequence actually counts backward from $Z$.
Fix: Confirm whether the direction of alphabet jumps is strictly positive or negative.
9. Assuming Strict Mathematical Uniqueness
Trap: Panicking when two plausible rules exist.
Fix: Check answer choices! In multiple-choice aptitude tests, only one valid rule will be among the options.
10. Selecting a Wrong Term Without Re-Verifying
Trap: Spotting an anomaly and picking it without verifying if replacing it restores all subsequent terms.
Fix: Perform the substitution and confirm the restored number links seamlessly to the next term.
Shortcuts & TBT Exam Playbook
Spend 10s checking differences, 10s checking ratios, 10s checking squares/cubes, and 15s checking alternating tracks. If no pattern emerges within 45 seconds, flag and skip. Never let a series question eat 3 minutes.
If you suspect an alternating rule or composite multiplier, plug the answer choices into the blank. Checking if an option satisfies both the predecessor and successor differences resolves the problem in seconds.
Commit to memory: Primes to 100 ($2, 3 \dots 97$), Squares to $32^2$ ($1024$), Cubes to $12^3$ ($1728$), Powers of 2 to $2^{10}$ ($1024$), Powers of 3 to $3^6$ ($729$), and Factorials to $7!$ ($5040$).
15 Verified Practice Questions
Find the missing number in the sequence: $17, 24, 31, 38, ?, 52$
Deduction: Differences: $24 - 17 = 7$; $31 - 24 = 7$; $38 - 31 = 7$.
This is an arithmetic progression with common difference $+7$.
Missing term: $38 + 7 = \mathbf{45}$.
Bridging check: $45 + 7 = 52$ (Matches!).
What is the next number in the sequence: $5, 15, 45, 135, ?$
Deduction: Multiplicative ratio: $15/5 = 3$; $45/15 = 3$; $135/45 = 3$.
Common ratio $r = 3$.
Next term: $135 \times 3 = \mathbf{405}$.
Find the next letter in the series: $D, G, J, M, P, ?$
Deduction: Positional ranks: $D(4), G(7), J(10), M(13), P(16)$.
Constant jump: $+3$.
Next position: $16 + 3 = 19 \implies \mathbf{S}$.
Find the next term in the sequence: $3, 6, 11, 18, 27, ?$
Deduction: First differences: $6-3 = 3$; $11-6 = 5$; $18-11 = 7$; $27-18 = 9$.
Differences are consecutive odd integers ($3, 5, 7, 9$). Next difference is $+11$.
Next term: $27 + 11 = \mathbf{38}$ (Also $n^2 + 2$ for $n=1..5$: $1+2=3, 4+2=6, 9+2=11, 16+2=18, 25+2=27 \implies 36+2=38$).
Determine the next term in the sequence: $0, 3, 8, 15, 24, ?$
Deduction: Form: $n^2 - 1$ for $n = 1, 2, 3, 4, 5$.
$1^2-1=0; 2^2-1=3; 3^2-1=8; 4^2-1=15; 5^2-1=24$.
Next term for $n=6$: $6^2 - 1 = 36 - 1 = \mathbf{35}$.
Find the next term in the series: $0, 7, 26, 63, 124, ?$
Deduction: Form: $n^3 - 1$ for $n = 1, 2, 3, 4, 5$.
$1^3-1=0; 2^3-1=7; 3^3-1=26; 4^3-1=63; 5^3-1=124$.
Next term for $n=6$: $6^3 - 1 = 216 - 1 = \mathbf{215}$.
What is the next number in the sequence: $2, 3, 5, 7, 11, 13, 17, ?$
Deduction: The sequence contains prime numbers in ascending order starting from 2.
The prime immediately following 17 is $\mathbf{19}$.
Determine the next term: $25, 10, 22, 13, 19, 16, 16, ?$
Deduction: Two interleaved series:
Odd positions: $25 \xrightarrow{-3} 22 \xrightarrow{-3} 19 \xrightarrow{-3} 16$.
Even positions: $10 \xrightarrow{+3} 13 \xrightarrow{+3} 16 \xrightarrow{+3} \mathbf{19}$.
Target is at position 8 (even track): $16 + 3 = \mathbf{19}$.
Find the next number in the sequence: $2, 7, 17, 32, 52, ?$
Deduction: $\Delta_1$: $7-2=5$; $17-7=10$; $32-17=15$; $52-32=20$.
Differences increase by $+5$ each time ($\Delta_2 = +5$).
Next difference: $20 + 5 = 25$.
Next term: $52 + 25 = \mathbf{77}$.
Find the next letter in the series: $A, C, F, J, O, ?$
Deduction: Positions: $A(1) \xrightarrow{+2} C(3) \xrightarrow{+3} F(6) \xrightarrow{+4} J(10) \xrightarrow{+5} O(15)$.
Next jump: $+6$.
$15 + 6 = 21 \implies \mathbf{U}$.
Find the next term in the series: $4, 9, 20, 43, 90, ?$
Deduction: Rule: $T_n = 2 T_{n-1} + n$:
$4 \times 2 + 1 = 9$
$9 \times 2 + 2 = 20$
$20 \times 2 + 3 = 43$
$43 \times 2 + 4 = 90$
Next: $90 \times 2 + 5 = 180 + 5 = \mathbf{185}$.
Identify the incorrect term in the sequence: $3, 7, 16, 35, 72, 153, 312$
Deduction: Generative law: $T_n = 2 T_{n-1} + n$:
$3 \times 2 + 1 = 7$
$7 \times 2 + 2 = 16$
$16 \times 2 + 3 = 35$
$35 \times 2 + 4 = \mathbf{74}$ (Given term is 72!)
Verification with 74: $74 \times 2 + 5 = 153$; $153 \times 2 + 6 = 312$.
Hence, $\mathbf{72}$ is incorrect; it should be 74.
What is the next term in the series: $W12D, T15G, Q18J, N21M, ?$
Deduction: Decompose into 3 parallel tracks:
• Letter 1: $W(23) \xrightarrow{-3} T(20) \xrightarrow{-3} Q(17) \xrightarrow{-3} N(14) \xrightarrow{-3} \mathbf{K(11)}$
• Number: $12 \xrightarrow{+3} 15 \xrightarrow{+3} 18 \xrightarrow{+3} 21 \xrightarrow{+3} \mathbf{24}$
• Letter 2: $D(4) \xrightarrow{+3} G(7) \xrightarrow{+3} J(10) \xrightarrow{+3} M(13) \xrightarrow{+3} \mathbf{P(16)}$
Synthesis: $\mathbf{K24P}$.
Find the next term in the sequence: $2, 6, 12, 20, 30, ?$
Deduction: These are pronic (oblong) numbers of the form $n(n+1)$:
$1 \times 2 = 2$; $2 \times 3 = 6$; $3 \times 4 = 12$; $4 \times 5 = 20$; $5 \times 6 = 30$.
For $n=6$: $6 \times 7 = \mathbf{42}$.
(Alternatively, differences: $+4, +6, +8, +10 \implies +12 \implies 30 + 12 = 42$).
Find the next term in the series: $1, 5, 14, 30, 55, 91, ?$
Deduction: First differences are consecutive perfect squares:
$5 - 1 = 4 = 2^2$
$14 - 5 = 9 = 3^2$
$30 - 14 = 16 = 4^2$
$55 - 30 = 25 = 5^2$
$91 - 55 = 36 = 6^2$
Next difference is $7^2 = 49$.
Next term: $91 + 49 = \mathbf{140}$.
Frequently Asked Questions
What is a number series?
A number series is an ordered sequence of numerical values generated by an underlying logical, mathematical, or structural rule. The goal is to identify that generative rule and apply it to deduce missing terms, find the next term, or isolate an incorrect term.
How do I start solving a missing-term series?
First, inspect the magnitude of change between adjacent terms. If numbers increase slowly or moderately, calculate first differences. If numbers grow rapidly by factors, test multiplicative ratios or powers (squares, cubes). If numbers oscillate up and down, separate the odd and even indexed terms into two independent alternating sub-series.
When should I use differences?
Use differences whenever the sequence exhibits steady, additive, or quadratic expansion where successive terms do not maintain a constant multiplication ratio. If first differences form an arithmetic progression, the second difference will be constant, indicating a quadratic pattern.
When should I use ratios?
Use ratios when terms grow exponentially or multiply dramatically (e.g., 4, 12, 36, 108). Divide each term by its predecessor. If the quotient is constant, it is a geometric series; if the multiplier increases by 1 each step (×2, ×3, ×4), it is a factorial-like expansion.
How do I recognise an alternating series?
An alternating series typically features numbers that fluctuate up and down (increase, decrease, increase, decrease) or has two distinct clusters of numbers. Separate terms at positions 1, 3, 5, 7 from terms at positions 2, 4, 6, 8 and analyze each track independently.
How do I solve alphabet series?
Immediately translate every letter into its standard numerical alphabet position (A=1, B=2, ... Z=26) using anchor mnemonics like EJOTY (5, 10, 15, 20, 25). Once in numeric form, analyze differences and wrap around from Z to A using modulo 26 arithmetic when required.
How do I solve alphanumeric series?
Deconstruct each cluster into independent letter and number components. Analyze the letter sequence and the numeric sequence on separate parallel tracks first, and then check whether the number represents the positional value or mathematical power of the letter.
How can I identify an incorrect term?
In a difference-based series, a single wrong number corrupts exactly two adjacent first differences. Locate the two corrupted differences; the number shared between those two intervals is the erroneous term. Replace it with the calculated correct value to verify that both differences and all remaining terms are restored.
Can a series have more than one valid continuation?
In pure mathematics, any finite sequence can technically be interpolated by infinitely many polynomials. However, in competitive aptitude tests, the intended answer is governed by the Principle of Parsimony (the simplest, most consistent rule requiring the fewest arbitrary assumptions that fits all provided terms).
How do I avoid overcomplicating a pattern?
Always test the basic foundational patterns first: constant difference, constant ratio, simple squares/cubes, and prime numbers. Never jump to complex polynomial formulas or obscure digit tricks until all elementary single-operation and two-track alternating rules have been tested and eliminated.
Related Concepts & What's Next
Advance next to Venn Diagrams — master 2-set and 3-set visual logical overlaps, "Only X" regions, survey analysis, and maximum-minimum overlap optimization for CAT and MBA CET.