Number and letter series

Placement essentials (aptitude, reasoning, verbal, communication, interviews, Excel, workplace) (All) · Logical reasoning · free study notes for exams, placements and interviews

In short

A series is an ordered set of numbers, letters or both that follows one hidden rule; the task is to find the rule and then supply a missing term, replace a wrong term, or continue the pattern. It is the fastest-scoring section in almost every Indian placement aptitude test (TCS NQT, Infosys, Wipro NLTH, Capgemini, Accenture, Cognizant), in bank and insurance exams (IBPS PO/Clerk, SBI, RBI Grade B, LIC AAO), and in SSC, RRB, CAT and MAT. University BCA/MCA/MBA and engineering "Quantitative Aptitude and Reasoning" papers under the AICTE model curriculum carry 5-10 marks on it. The skill tested is pattern recognition under time pressure - typically 30-45 seconds per question - so the method matters more than the arithmetic.

Explained

Question formats you will meet

Missing term: one blank anywhere in the series - 3, 7, 15, 31, __, 127. Mostly the last or middle term.
Wrong term (odd one out): every term except one obeys the rule - find and correct it. Common in SBI PO and RRB NTPC.
Continue the series: give the next one or two terms.
Two missing terms: harder; verify each against the same rule - or, if the series is interleaved, against the rule of its own sub-series.
Series as a question stem: "what should come in place of the question mark" with the series given in a table or as a sequence of letter pairs.
Alphanumeric / mixed series: a string such as A 4 % P 7 B @ 9 K with questions on "how many letters are immediately preceded by a digit" - solved by positional scanning, not by a rule.

Number series: the standard rule families

Arithmetic (constant difference): 7, 12, 17, 22 - difference +5. General term aₙ = a + (n−1)d.
Geometric (constant ratio): 3, 12, 48, 192 - ratio ×4. aₙ = a·rⁿ⁻¹.
Difference of differences (second-order): 2, 5, 11, 20, 32 - first differences 3, 6, 9, 12 (an AP), so the pattern lives one level down. If the second differences are constant, the series is quadratic in n.
Multiply and add / mixed operation: 3, 7, 15, 31, 63 - each term ×2 + 1. Also ×3 − 2, ×2 − 1, ×1.5 + 0.5 types.
Squares and cubes: 1, 4, 9, 16, 25; 1, 8, 27, 64; or shifted forms n²+1 (2, 5, 10, 17, 26), n²−1 (0, 3, 8, 15, 24), n³+n (2, 10, 30, 68).
Products of consecutive integers: 2, 6, 12, 20, 30 = n(n+1); 6, 24, 60, 120 = n(n+1)(n+2).
Prime-based: 2, 3, 5, 7, 11, 13 (primes); 4, 9, 25, 49, 121 (squares of primes); 2, 6, 12, 20 can also be read as sums.
Factorial based: 1, 2, 6, 24, 120 = n!; or 2, 3, 7, 25, 121 built as n!·k + something.
Fibonacci / recursive: each term = sum (or difference, or sum × constant) of the previous two: 1, 1, 2, 3, 5, 8; or 2, 3, 5, 8, 13; or 4, 7, 11, 18, 29.
Alternate (two interleaved series): 2, 9, 4, 12, 6, 15 - odd positions 2, 4, 6 (+2) and even positions 9, 12, 15 (+3). Very common in bank exams.
Increasing increments of a pattern: 10, 11, 13, 16, 20, 25 - add 1, 2, 3, 4, 5. Or ×1+1, ×2+2, ×3+3.
Division / fraction series: 720, 120, 24, 6, 2 - divide by 6, 5, 4, 3, 2. Decreasing-divisor series are a favourite trick.
Decimal and fraction series: 1/2, 3/4, 5/6, 7/8 - numerators are odd numbers, denominators even; treat numerator and denominator as separate series.
Digit-operation series: 21, 32, 54, 87 - add the sum of digits of the previous term (21+3=24? no) - here each term adds 11, 22, 33; genuine digit series exist, e.g. 14, 19, 29, 40 built from digit sums, so test digit sums only after the standard families fail.

A fixed procedure for number series

Write the differences between consecutive terms in a row below the series. If they are constant, you are done; if they form a recognisable pattern (AP, squares, ×2), use it.
If differences grow fast, test ratios. A roughly constant ratio means GP or "×k ± c".
If the ratio drifts upward, suspect a changing multiplier: ×1, ×2, ×3 or ×2+1, ×3+1.
Check against known lists - squares up to 30² = 900, cubes up to 15³ = 3,375, primes up to 100, factorials up to 7! = 5,040. Memorising these is what makes 30-second solving possible.
If terms rise then fall, or alternate in size, split the series into odd-position and even-position terms and solve each separately.
For a wrong-number question, assume the majority is right. Build the rule from three or four consecutive terms, then find the single term that breaks it; state the corrected value.
Verify the rule on every given term, not just the two next to the blank - this is the single biggest source of marks lost.

Letter series: the one thing you must memorise

The 26 positions: A1 B2 C3 D4 E5 F6 G7 H8 I9 J10 K11 L12 M13 N14 O15 P16 Q17 R18 S19 T20 U21 V22 W23 X24 Y25 Z26.
EJOTY (5, 10, 15, 20, 25) is the standard memory aid for fifth letters; CFILORUX marks every third letter.
Opposite-letter rule: the pair position sums to 27 - A↔Z, B↔Y, C↔X, M↔N. So the opposite of the k-th letter is the (27−k)-th. Memory aids: AZ, BY, CX, DW, EV, FU, GT, HS, IR, JQ, KP, LO, MN.
Types of letter series:
- Constant skip: A, D, G, J (+3); Z, W, T, Q (−3).
- Increasing skip: A, B, D, G, K (+1, +2, +3, +4).
- Two-letter groups: AZ, BY, CX - first letter forward, second backward.
- Three-letter groups: ABD, CDF, EFH - each group shifted by +2 from the previous group.
- Letter with repeated letters: AABBCC, or ABABC patterns.
- Cyclic (modulo 26): after Z, wrap to A. Position 29 → 29 − 26 = 3 = C.
Alphanumeric series: letters mixed with numbers - convert letters to positions, solve as a number series, convert back. Example: C, F, J, O, U → 3, 6, 10, 15, 21 → differences 3, 4, 5, 6 → next difference 7 → 28 → 28 − 26 = 2 = B.
Reverse alphabet positions: Z1 Y2 X3 ... useful when a series counts down; reverse position = 27 − forward position.

Series within a quadratic or formula frame (the exam-proof method)

If the second differences are constant and equal to 2a, the series is aₙ = an² + bn + c. Solve for a, b, c from three terms. This converts guessing into algebra and is the method that university papers and GATE-style aptitude sections expect when they ask you to "find the general term".
For a geometric series the n-th term is a·rⁿ⁻¹; the sum of n terms is a(rⁿ−1)/(r−1) for r ≠ 1 - occasionally asked as "find the sum of the first six terms of the series".
Recursive definitions are written as aₙ = f(aₙ₋₁) or aₙ = f(aₙ₋₁, aₙ₋₂); stating the recurrence is a complete answer in a written exam.

Advantages, limits and exam strategy

Advantages: no formula sheet needed, no lengthy calculation, high accuracy achievable, and the same skill transfers to coding-decoding and analogies.
Limitation - ambiguity: more than one rule can fit a short series (2, 4, 8 is ×2 giving 16, or differences 2, 4, 6 giving 14). Choose the simplest rule that fits all given terms, and if options are given, the one that matches an option.
Limitation - time: never spend over 60 seconds. Mark and move; in sectional-cut-off exams (IBPS, SBI) accuracy beats attempts because of negative marking (typically 0.25 per wrong answer - check the current notification, as marking schemes change).
Back-solving from options is legitimate and fast: substitute each option into your candidate rule.
For wrong-number questions, writing out the corrected full series takes five seconds and removes doubt.

Key points

Series - an ordered sequence governed by one rule; the rule must explain every given term.
First differences - subtract consecutive terms; the first and most productive check.
Second differences constant - the series is quadratic, aₙ = an² + bn + c.
Geometric series - constant ratio r; aₙ = a·rⁿ⁻¹.
Mixed operation - ×k ± c pattern; detect it when ratios are near-constant but not exact.
Alternate series - two interleaved series; split by odd and even positions.
Fibonacci type - each term from the previous two; test aₙ = aₙ₋₁ + aₙ₋₂ early when growth is moderate.
Letter positions - A1 to Z26 must be instant; EJOTY = 5, 10, 15, 20, 25.
Opposite letter - positions sum to 27; opposite of k is 27 − k.
Cyclic wrap - position > 26 reduces modulo 26 (position 27 → A).
Reverse position - 27 − forward position; used in descending letter series.
Memorise lists - squares to 900, cubes to 3,375, primes under 100, factorials to 5,040.
Wrong-number rule - the majority defines the pattern; one term is the outlier.
Simplest rule wins - avoid elaborate rules when a plain one fits.
Verify on all terms - a rule checked on only two terms is the commonest error.
Decreasing divisor - 720, 120, 24, 6: ÷6, ÷5, ÷4, ÷3 - watch for falling series.
Time discipline - 30-45 seconds per question; abandon at 60.
Negative marking - usually 0.25 marks per wrong answer in bank exams (IBPS, SBI) and 0.50 in SSC CGL/CHSL Tier 1; confirm from the current notification.

Formulas and examples

Arithmetic progression: aₙ = a + (n − 1)d, where a = first term, d = common difference, n = term number (all dimensionless)
Sum of n AP terms: Sₙ = n/2 [2a + (n − 1)d]
Geometric progression: aₙ = a·rⁿ⁻¹; Sₙ = a(rⁿ − 1)/(r − 1) for r ≠ 1
Quadratic series: aₙ = an² + bn + c, where 2a = constant second difference
Letter position: position of a letter L = its rank in A..Z (A = 1 ... Z = 26); position of the opposite letter = 27 − position(L)
Cyclic shift: new position = ((old position + shift − 1) mod 26) + 1
Recursive (Fibonacci type): aₙ = aₙ₋₁ + aₙ₋₂, with a₁ and a₂ given
Sum of first n natural numbers = n(n+1)/2; sum of squares = n(n+1)(2n+1)/6; sum of cubes = [n(n+1)/2]²

Example 1 - multiply-and-add series. Find the missing term: 3, 7, 15, 31, __, 127.

Differences: 4, 8, 16, … - they double, so a plain AP is out.
Test ×2 + 1: 3×2+1 = 7 ✔; 7×2+1 = 15 ✔; 15×2+1 = 31 ✔.
So missing term = 31×2+1 = 63; check 63×2+1 = 127 ✔.
General form: aₙ = 2ⁿ⁺¹ − 1 (n = 1 gives 3, n = 6 gives 127).

Example 2 - quadratic series by second differences. Continue: 2, 5, 11, 20, 32, __.

First differences: 3, 6, 9, 12 → second differences: 3, 3, 3 (constant).
Next first difference = 15, so next term = 32 + 15 = 47.
Algebraic check: 2a = 3 → a = 1.5. Using a₁ = 2 and a₂ = 5: 1.5 + b + c = 2 and 6 + 2b + c = 5 → subtract: 4.5 + b = 3 → b = −1.5, then c = 2. So aₙ = 1.5n² − 1.5n + 2. For n = 6: 1.5(36) − 1.5(6) + 2 = 54 − 9 + 2 = 47 ✔.

Example 3 - wrong number. One term is wrong: 6, 12, 36, 150, 720, 4320.

Test successive multipliers: 12/6 = 2, 36/12 = 3, 150/36 = 4.17, 720/150 = 4.8, 4320/720 = 6.
The intended pattern is ×2, ×3, ×4, ×5, ×6. Then the fourth term should be 36 × 4 = 144, not 150.
Verify onward: 144 × 5 = 720 ✔; 720 × 6 = 4320 ✔. So 150 is wrong; it should be 144.

Example 4 - letter series with increasing skip. Complete: C, F, J, O, __.

Positions: C = 3, F = 6, J = 10, O = 15.
Differences: 3, 4, 5 → next difference 6 → position 15 + 6 = 21 → U.
Continuing: next difference 7 → 28 → 28 − 26 = 2 → B (cyclic wrap).

Example 5 - alternate (interleaved) series and letter pairs.

Numbers: 5, 11, 9, 14, 13, 17, __, __. Odd positions 5, 9, 13 (+4) → next 17. Even positions 11, 14, 17 (+3) → next 20. Answers: 17 and 20.
Letter pairs: AZ, CX, EV, GT, __. First letters A, C, E, G = 1, 3, 5, 7 (+2) → 9 = I. Second letters Z, X, V, T = 26, 24, 22, 20 (−2) → 18 = R. Answer: IR - which is also the opposite-letter pair since 9 + 18 = 27 ✔.

Clue in the series	Most likely rule	First check to run
Constant difference	AP	subtract consecutive terms
Differences form an AP	Quadratic / n² type	second differences
Terms roughly double or triple	GP or ×k ± c	divide consecutive terms
Growth faster than GP	n!, nⁿ, or ×n changing	compare with 1, 2, 6, 24, 120
Terms 1, 4, 9, 16 nearby	n² ± c	subtract nearest square
Terms fall rapidly	decreasing divisors	divide consecutive terms
Values zig-zag up and down	two interleaved series	split odd/even positions
Moderate growth, no clean ratio	Fibonacci type	add previous two terms
Letters given	convert to positions 1-26	then treat as number series
Pairs of letters	separate forward/backward series	check sum = 27 for opposites

Letter-position ready reference
A  B  C  D  E  F  G  H  I  J  K  L  M
1  2  3  4  5  6  7  8  9 10 11 12 13
N  O  P  Q  R  S  T  U  V  W  X  Y  Z
14 15 16 17 18 19 20 21 22 23 24 25 26
Opposites (sum 27): A-Z B-Y C-X D-W E-V F-U G-T H-S I-R J-Q K-P L-O M-N

Where it is used

IT service hiring: TCS NQT, Infosys, Wipro NLTH, Capgemini, Accenture, Cognizant and LTIMindtree aptitude rounds all carry number and letter series in the reasoning section.
Banking, insurance and PSU exams: IBPS PO/Clerk/RRB, SBI PO/Clerk, RBI Grade B, LIC AAO, NABARD - number series is a dedicated 5-question block in the quantitative paper, plus alphanumeric series in reasoning.
Government and defence recruitment: SSC CGL/CHSL, RRB NTPC and Group D, CAPF, CDS, AFCAT, state PSC prelims and EPFO/ESIC exams.
Management entrance and university papers: CAT, XAT, MAT, CMAT, NMAT logical reasoning; BCA/MCA/MBA and AICTE-curriculum engineering "Quantitative Aptitude and Reasoning" courses.
On the job: the same pattern-recognition skill underlies spotting a trend or anomaly in sales, attendance or defect data, checking sequence integrity in invoice or requisition numbers, and writing loop logic and recurrences in programming.
Programming interviews: sequence generation questions (Fibonacci, primes, n-th term of a recurrence) in coding rounds are the same ideas expressed as code.

Interview questions

Q: What comes next in 3, 7, 15, 31, 63? A: 127 - each term is the previous one doubled plus 1, that is aₙ = 2ⁿ⁺¹ − 1.
Q: How do you decide whether a series is arithmetic or geometric? A: Take consecutive differences first; if they are constant it is arithmetic, and if the ratios are constant instead it is geometric.
Q: What does a constant second difference tell you? A: The series is quadratic, aₙ = an² + bn + c, with the constant second difference equal to 2a.
Q: Complete the letter series B, E, I, N. A: T - positions 2, 5, 9, 14 with differences 3, 4, 5, so the next difference is 6, giving position 20 = T.
Q: Which letter is opposite to H in the alphabet? A: S, because the positions of an opposite pair add to 27 and 8 + 19 = 27.
Q: Find the wrong term in 2, 6, 12, 20, 30, 44. A: 44 should be 42 - the pattern is n(n+1): 2, 6, 12, 20, 30, 42, with differences 4, 6, 8, 10, 12.
Q: 1, 1, 2, 3, 5, 8, 13 - name and extend it. A: The Fibonacci series, each term the sum of the previous two; the next terms are 21 and 34.
Q: What is the 15th letter from the right end of the alphabet? A: L - the 15th from the right is the (27 − 15) = 12th from the left, which is L.
Q: A series can fit two rules - how do you choose? A: Pick the simplest rule that satisfies every given term, and when options are provided, the one whose answer appears among the options.
Q: 720, 120, 24, 6, 2 - what is the next term? A: 1 - the divisors fall as 6, 5, 4, 3, then 2, so 2 ÷ 2 = 1; the series is 6!, 5!, 4!, 3!, 2!, 1!.
Q: How would you code a general "next term" finder? A: You cannot in general - infinitely many rules fit a finite sequence; practically you test a fixed library of patterns (AP, GP, ×k±c, polynomial fit, recurrences) and report the simplest match.
Q: Why do banks test number series rather than heavy calculation? A: It measures speed of pattern recognition and numerical sense in 30-40 seconds, which correlates with the mental arithmetic and anomaly-spotting a clerical or officer role needs.

Common mistakes

Fixing the rule from only the two terms beside the blank, then failing on an earlier term.
Missing an interleaved (alternate) series when values move up and down, and forcing a single rule on it.
Not writing the differences down - trying to spot the pattern purely mentally and losing time.
Forgetting the cyclic wrap in letter series, so position 29 is treated as impossible instead of C.
Confusing forward and reverse letter positions, or the "from the right end" count (use 27 − k).
In wrong-number questions, changing the first or last term because it "looks odd" instead of letting the majority of terms define the rule.
Spending two or three minutes on one ambiguous series and losing four easy questions elsewhere.