Number & Letter Series
Number & Letter Series is a frequently tested area in CUET General Test. Work through these free NTA-style sample questions with full answers and explanations, then attempt all 40 in a timed practice test to build exam-day speed.
Snapshot
- Number & Letter Series shows a sequence with a missing term (or one wrong term) and asks you to find it. The whole skill is spotting the pattern, and the pattern almost always lives in the differences, in a multiplier, or in squares, cubes and primes.
- Letter series run on alphabet positions (A = 1 ... Z = 26), so reading positions fast is half the battle.
- CUET uses several question shapes: next term, missing middle term, wrong term, alternating series, letter series, alpha-numeric series and repeating-pattern letter series.
- Write the differences first. If they do not settle it, check second differences, multipliers, alternation and special sequences, in that order. Exam reality: +5 for a correct answer and −1 for a wrong one.
Part 1 — The method: a fixed order of attack
Written and reviewed by Devendra Dubey, Mathematics & Commerce faculty at UniDrill (M.Sc. Mathematics, Osmania University · 10 years teaching). Prepared for the CUET UG General Test.
The commonest reason students fail at series is guessing a rule from the first two terms. A pattern needs to hold across the whole series, so test it on every gap. Follow this order every time.
| Step | What to check | Signal |
|---|---|---|
| 1. Look at the size of the jumps | Slow growth means add or subtract; fast growth means multiply or powers | Terms barely change vs. terms explode |
| 2. Write the first differences | Subtract each term from the next | Constant gives arithmetic; a pattern in the gaps gives a second level |
| 3. Write the second differences | Differences of the differences | Constant second difference signals a quadratic such as n² + n |
| 4. Try a multiplier | Divide each term by the one before | A constant ratio gives geometric; ratios 1, 2, 3, 4 give factorial-like growth |
| 5. Compare with special numbers | Squares, cubes, primes, Fibonacci | Terms sit near n², n³ or 2ⁿ |
| 6. Try alternation | Split into odd-placed and even-placed terms | Two slow patterns hidden in one fast one |
| 7. Try a two-step rule | ×a then +b, or +1², +2², +3² | Differences themselves form a pattern |
Verify on every gap. If your rule predicts the third term correctly but misses the fourth, it is the wrong rule. A good rule explains all given terms, and the answer is then forced.
Part 2 — The full menu of series questions
| Type | What you see | Approach |
|---|---|---|
| Next term | 3, 7, 13, 21, 31, ? | Differences, then continue |
| Missing middle term | 2, 6, ?, 20, 30, 42 | Find the rule from the other gaps, then fill |
| Wrong term | 2, 5, 10, 17, 27, 37 | Establish the rule from reliable terms; find the one that violates it |
| Alternating series | 1, 4, 27, 16, 125, 36, ? | Split into two series |
| Two-step rule | 3, 4, 10, 33, 136, ? | Each step uses a changing multiplier and adder |
| Special number series | 2, 3, 5, 7, 11, 13, ? | Primes, squares, cubes, Fibonacci |
| Letter series | A, C, F, J, ? | Convert to positions, find the gaps |
| Alternating letter series | B, Y, D, W, F, U, ? | Odd and even places run in different directions |
| Alpha-numeric series | A1, C3, E5, G7, ? | Letter and number each follow their own rule |
| Repeating-pattern letters | ab_cab_c_bbc | Find the repeating block and fill the blanks |
Part 3 — Number series: the engines
Know these engines by sight. Most CUET number series are one of them.
| Engine | Example | Rule | Next |
|---|---|---|---|
| Arithmetic | 7, 12, 17, 22 | +5 | 27 |
| Growing differences | 3, 7, 13, 21, 31 | Gaps 4, 6, 8, 10 | 43 |
| Second-level constant | 2, 6, 12, 20, 30 | n(n + 1); gaps 4, 6, 8, 10 | 42 |
| Geometric | 3, 9, 27, 81 | ×3 | 243 |
| Squares | 1, 4, 9, 16, 25 | n² | 36 |
| Cubes | 1, 8, 27, 64 | n³ | 125 |
| Squares ± k | 2, 5, 10, 17, 26 | n² + 1 | 37 |
| Cubes ± k | 0, 7, 26, 63, 124 | n³ − 1 | 215 |
| Primes | 2, 3, 5, 7, 11 | Primes | 13 |
| Fibonacci type | 2, 3, 5, 8, 13, 21 | Sum of previous two | 34 |
| Lucas type | 1, 3, 4, 7, 11, 18 | Sum of previous two | 29 |
| Factorial-like | 1, 2, 6, 24, 120 | ×1, ×2, ×3, ×4, ×5 | 720 |
| Multiply and add | 5, 11, 23, 47 | ×2 + 1 | 95 |
| Changing multiplier and adder | 3, 4, 10, 33, 136 | ×1+1, ×2+2, ×3+3, ×4+4 | 685 |
| Powers of 2 | 2, 4, 8, 16 | ×2 | 32 |
| Descending squares minus one | 120, 99, 80, 63, 48 | n² − 1 downward | 35 |
How the second-difference test works. Take 6, 11, 21, 36, 56. First differences are 5, 10, 15, 20 and these themselves rise by 5, so the second differences are constant. The next first difference is 25 and the next term is 56 + 25 = 81. Whenever the first differences are not constant but are themselves regular, you are in this family.
When the numbers jump fast. If a series goes 2, 6, 18, 54, differences of 4, 12, 36 are not helpful. The ratio is 3 every time, so multiply instead. Test ratio before you go deep into differences when the growth looks explosive.
Part 4 — Letter series
Convert letters into positions and run the same differences method on the numbers. Anchor letters help: A = 1, E = 5, J = 10, O = 15, T = 20, Y = 25 (remember "EJOTY").
| Pattern | Example | Reading |
|---|---|---|
| Fixed gap | A, C, E, G | +2 each (1, 3, 5, 7) |
| Backward gap | Z, X, V, T | −2 each |
| Growing gap | A, C, F, J, O | Gaps 2, 3, 4, 5: positions 1, 3, 6, 10, 15 |
| Alternating directions | B, Y, D, W, F, U | Odd places forward (B, D, F), even places backward (Y, W, U) |
| Mirror steps | A, Z, B, Y, C, X | Alternate letters from each end |
| Wrap-around | W, Z, C, F | +3 each; Z + 3 wraps to C |
| Grouped letters | AB, CD, EF | The pair moves as a block |
Wrap-around. Positions continue after 26: position 27 is A, 28 is B and so on. So Y + 4 lands at position 29, which is C.
Part 5 — Alpha-numeric series
Treat the letters and the numbers as two separate series, and look for each rule in turn.
| Example | Letter rule | Number rule | Next |
|---|---|---|---|
| A1, C3, E5, G7, ? | +2 | +2 | I9 |
| B2, D8, F18, H32, ? | +2 | Position squared × 2? (2 = 2×1, 8 = 2×4, 18 = 2×9, 32 = 2×16) | J50 |
| A2, B4, C8, D16, ? | +1 | ×2 | E32 |
The second example is more advanced. The letters B, D, F, H sit at positions 2, 4, 6, 8, and the numbers are 2, 8, 18, 32 = 2 × (1², 2², 3², 4²). The next term uses 2 × 5² = 50, and the letter J.
Part 6 — Repeating-pattern letter series
A series like ab_cab_c_bbc is built from a block that repeats. Find the block from the unbroken part. Here the block is abbc (a, b, b, c), written three times: abbc abbc abbc. Reading the question against it: a b [b] c a b [b] c [a] b b c. The missing letters are b, b, a.
Method: (1) guess the block length from where the first letter returns, (2) write the full repeating string below the question, (3) read off the blanks. Check that every given letter matches the string.
Part 7 — Worked examples with full reasoning
1. 3, 7, 13, 21, 31, ? First differences: 4, 6, 8, 10. They rise by 2, so the next is 12. The answer is 31 + 12 = 43.
2. 5, 11, 23, 47, ? Differences 6, 12, 24 are doubling, which hints at ×2 + something. Test: 5 × 2 + 1 = 11, 11 × 2 + 1 = 23, 23 × 2 + 1 = 47. Then 47 × 2 + 1 = 95.
3. 2, 6, 12, 20, 30, ? Differences: 4, 6, 8, 10 (next 12). 30 + 12 = 42. Check with the formula n(n + 1): 1×2 = 2, 2×3 = 6, 3×4 = 12, 4×5 = 20, 5×6 = 30, 6×7 = 42.
4. 1, 2, 6, 24, 120, ? Ratios: 2, 3, 4, 5. The next ratio is 6, so the answer is 120 × 6 = 720.
5. 2, 3, 5, 8, 13, 21, ? Each term is the sum of the two before: 2 + 3 = 5, 3 + 5 = 8, 5 + 8 = 13, 8 + 13 = 21. Next: 13 + 21 = 34.
6. 1, 4, 27, 16, 125, 36, ? The series jumps up and down, so split it. Odd places: 1, 27, 125 are 1³, 3³, 5³. Even places: 4, 16, 36 are 2², 4², 6². The seventh term is an odd place, so it is 7³ = 343.
7. 3, 4, 10, 33, 136, ? Test a changing multiplier and adder. 3 × 1 + 1 = 4, 4 × 2 + 2 = 10, 10 × 3 + 3 = 33, 33 × 4 + 4 = 136. Next: 136 × 5 + 5 = 685.
8. 120, 99, 80, 63, 48, ? Differences: −21, −19, −17, −15. They are rising by 2, so the next is −13. The answer is 48 − 13 = 35. As a check, the terms are 11² − 1, 10² − 1, 9² − 1, 8² − 1, 7² − 1, so the next is 6² − 1 = 35.
9. 6, 11, 21, 36, 56, ? First differences: 5, 10, 15, 20, rising by 5. The next difference is 25. The answer is 56 + 25 = 81.
10. A, C, F, J, ? Positions: 1, 3, 6, 10. Differences: 2, 3, 4, so the next is 5. Position 10 + 5 = 15 is O.
11. Z, W, T, Q, ? Positions: 26, 23, 20, 17. Each step is −3, so the next is 14, which is N.
12. B, Y, D, W, F, U, ? Split into odd places (B, D, F) and even places (Y, W, U). Odd places go +2 (positions 2, 4, 6), even places go −2 (25, 23, 21). The seventh term continues the odd places: position 8, which is H.
13. Find the wrong term: 1, 3, 9, 28, 81, 243. Ratios of 3 fit the first term onward: 1, 3, 9, 27, 81, 243. The term 28 should be 27. The wrong term is 28.
14. Find the wrong term: 2, 5, 10, 17, 27, 37. The rule n² + 1 gives 2, 5, 10, 17, 26, 37. The term 27 should be 26. The wrong term is 27.
15. ab_cab_c_bbc. The block is abbc. The three blanks are b, b, a.
16. A1, C3, E5, G7, ? Letters are +2 (A, C, E, G), so the next letter is I. Numbers are +2, so the next is 9. The answer is I9.
Part 8 — Common traps
| Trap | Why it is tempting | The truth |
|---|---|---|
| Forcing one rule on an alternating series | The series looks like it should have one rule | If the differences jump up and down, split the series by place |
| Stopping at first differences | They look irregular | Take the second differences before you give up |
| Fitting a rule to two terms only | Two terms fit many rules | A rule must hold on every gap |
| Off-by-one in letter positions | A feels like 0 | A is 1, Z is 26 |
| Letter gaps by eye | The alphabet looks evenly spaced | Convert to positions every time |
| Ignoring wrap-around | You stop counting at Z | Position 27 is A again |
| Missing squares and cubes | Numbers look random | Compare the terms with n² and n³ |
| Choosing the wrong term in a "wrong term" question | You pick the first odd looking number | Establish the rule from several terms and see which one fails |
| Ratio versus difference | Fast growth looks arithmetic | If growth is explosive, check ratios first |
Part 9 — Rapid-fire practice
Write the differences or ratios first. Then check the answer.
| No. | Series | Answer | Rule |
|---|---|---|---|
| 1 | 4, 9, 16, 25, 36, ? | 49 | Squares |
| 2 | 2, 4, 8, 16, ? | 32 | ×2 |
| 3 | 100, 95, 90, 85, ? | 80 | −5 |
| 4 | 1, 8, 27, 64, ? | 125 | Cubes |
| 5 | 3, 6, 11, 18, 27, ? | 38 | n² + 2 (gaps 3, 5, 7, 9, 11) |
| 6 | 2, 3, 5, 7, 11, ? | 13 | Primes |
| 7 | 1, 2, 4, 7, 11, 16, ? | 22 | Gaps 1, 2, 3, 4, 5, 6 |
| 8 | 2, 12, 36, 80, 150, ? | 252 | n³ + n² |
| 9 | B, D, G, K, ? | P | Gaps 2, 3, 4, 5 (11 + 5 = 16) |
| 10 | A, E, I, M, ? | Q | +4 |
| 11 | Z, X, U, Q, ? | L | Gaps −2, −3, −4, −5 (17 − 5 = 12) |
| 12 | AZ, BY, CX, DW, ? | EV | Mirror pairs advancing |
| 13 | 4, 6, 9, 13, 18, ? | 24 | Gaps 2, 3, 4, 5, 6 |
| 14 | Wrong term: 1, 4, 9, 16, 24, 36 | 24 | Squares; 24 should be 25 |
| 15 | 0, 3, 8, 15, 24, ? | 35 | n² − 1 |
Part 10 — A timed-practice routine
| Day | Task | Target |
|---|---|---|
| 1 | Memorise squares to 20, cubes to 10, primes to 50 and the EJOTY anchors | All lists from memory |
| 2 | 20 number series, writing first and second differences | Rule named for every question |
| 3 | 15 letter series with positions written under each letter | No eye-balling the alphabet |
| 4 | 10 alternating, wrong-term and alpha-numeric series | Learn to split and test |
| 5 | Mixed set of 20 | 15 minutes, about 45 seconds each |
| 6 | Redo every wrong one; write which engine it used | List the engines you miss |
| 7 | Timed set of 20 and the 30 practice questions attached to this page | 12 minutes for the set |
If you cannot find a rule in 40 seconds, move on. A series you cannot crack is worth less than two you can.
One-line revision: write the differences first, check second differences and ratios for fast jumps, split alternating series by place, and read letters by position using the EJOTY anchors.
Part 11 — Where this connects
Series is the engine behind number and letter Analogy and the "odd one out" patterns in Classification. Letter positions power Coding-Decoding, while squares, cubes and primes link to Arithmetic. Figure-based series appear in Non-Verbal Reasoning.
Practice questions
Now test yourself. 8 free sample questions with explanations. 32 more in the timed practice test.
Q1. Find the next term: $1, 3, 6, 10, 15, 21, ?$
▸ Show answer & explanation
Answer: C
These are triangular numbers; differences are $2,3,4,5,6,7$, so $21+7=28$.
Q2. Find the missing term: $5, 7, 12, 19, 31, 50, ?$
▸ Show answer & explanation
Answer: C
From the third term, each term is the sum of the two preceding terms: $19+31=50$ and $31+50=81$.
Q3. Find the next term: $10, 100, 200, 310, 430, ?$
▸ Show answer & explanation
Answer: C
Differences increase by $10$ each step: $90,100,110,120,130$; so $430+130=560$.
Q4. Find the next term in the series: $2, 3, 5, 7, 11, 13, ?$
▸ Show answer & explanation
Answer: C
The series lists consecutive prime numbers; after $13$ the next prime is $17$.
Q5. Find the next term in the series: $625, 125, 25, 5, ?$
▸ Show answer & explanation
Answer: B
Each term is divided by $5$: $5\div5=1$.
Q6. Find the next term in the series: $50, 45, 40, 35, 30, ?$
▸ Show answer & explanation
Answer: C
Each term decreases by $5$: $30-5=25$.
Q7. Find the next term: $11, 13, 17, 19, 23, 25, ?$
▸ Show answer & explanation
Answer: C
Differences alternate $+2,+4,+2,+4,+2,+4$: $25+4=29$.
Q8. Find the next term in the series: $1, 4, 9, 16, 25, ?$
▸ Show answer & explanation
Answer: C
These are perfect squares $1^2,2^2,3^2,4^2,5^2$; next is $6^2=36$.
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