Sequences — SSC CGL Reasoning Notes

Concepts and rules

A series follows one rule from term to term. You may be asked for the next term, a missing term, or the wrong term. The rule must fit every given term, not just the first few.

Difference series: a constant difference (3, 7, 11, 15) is an arithmetic progression. If the differences change, write them as a new series: 2, 6, 12, 20, 30 has differences 4, 6, 8, 10, which rise by 2.

Ratio series: each term is the previous one × k, or × k ± c. 5, 11, 23, 47 follows × 2 + 1. The multiplier itself can change (× 1, × 2, × 3).

Power series: n², n³, n² ± 1, n³ ± 1, n(n + 1). For example 3, 8, 15, 24, 35 is n² − 1 for n = 2, 3, 4, 5, 6, and 2, 9, 28, 65 is n³ + 1.

Interleaved series: the terms in odd places follow one rule and the terms in even places another. In 4, 7, 8, 14, 16, 21, … the odd places double (4, 8, 16) and the even places add 7 (7, 14, 21).

Letter series: change letters to positions (A=1 … Z=26, anchors E=5, J=10, O=15, T=20, Y=25) and use the same tools. Gaps may be constant (+3) or growing (+3, +4, +5), and they wrap after Z.

Letter clusters: in AZ, CX, FU, … treat each position as its own series: first letters A, C, F (+2, +3) and second letters Z, X, U (−2, −3).

Wrong term: the rule fits all terms but one. Correct that term and check that every other term still fits; if 2 terms fail, you have the wrong rule.

Method and shortcuts

Step 1: Look at the growth. Slow and steady suggests differences; roughly doubling suggests × 2 ± c; values near 4, 9, 16, 25 or 8, 27, 64 suggest powers; a zig-zag suggests 2 interleaved series.

Step 2: Write the first differences under the series; if they are not constant, write the second differences or the ratios.

Step 3: State the rule in words, test it on every given term, and only then compute the answer.

Shortcut: subtract the nearest square or cube from each term; a constant remainder such as −1 or +1 exposes n² ± 1 or n³ ± 1 at once.

Shortcut: if big and small jumps alternate, split the series into odd and even places before doing anything else.

Shortcut for letters running backwards: use reverse positions (27 − n), which turns a backward series into a forward one.

When it fails: if neither differences nor ratios settle, try a mixed rule such as × 2 + 1, × 2 + 2, × 2 + 3 or alternating + and ×.

Example 1: Constant difference

Question: 3, 7, 11, 15, ?

Solution: Each term is 4 more than the previous one, so the next term is 15 + 4 = 19.

Answer: 19

Example 2: Differences in a pattern

Question: 2, 6, 12, 20, 30, ?

Solution: Differences are 4, 6, 8, 10, so the next difference is 12 and the term is 30 + 12 = 42. The same series is n(n + 1): 1 × 2, 2 × 3, … , 6 × 7 = 42.

Answer: 42

Example 3: Letter gaps

Question: B, E, H, K, ?

Solution: B(2), E(5), H(8), K(11): the gap is +3 each time, so the next letter is 11 + 3 = 14, which is N.

Answer: N

Example 4: Multiply and add

Question: 5, 11, 23, 47, ? (a) 94 (b) 95 (c) 96 (d) 93

Solution: Each term roughly doubles: 5 × 2 + 1 = 11, 11 × 2 + 1 = 23, 23 × 2 + 1 = 47. So the next term is 47 × 2 + 1 = 95.

Check with differences: 6, 12, 24 double each time, so the next difference is 48 and 47 + 48 = 95 again.

Answer: (b) 95

Example 5: Letter clusters

Question: AZ, CX, FU, JQ, ? (a) PL (b) OK (c) OL (d) NL

Solution: First letters A(1), C(3), F(6), J(10): gaps +2, +3, +4, so next +5 gives 15 = O.

Second letters Z(26), X(24), U(21), Q(17): gaps −2, −3, −4, so next −5 gives 12 = L. Option (a) takes +6 for the first letter and (b) takes −6 for the second.

Answer: (c) OL

Example 6: Wrong term

Question: Find the wrong term: 3, 8, 15, 24, 34, 48, 63. (a) 8 (b) 24 (c) 48 (d) 34

Solution: Compare with squares: 4 − 1 = 3, 9 − 1 = 8, 16 − 1 = 15, 25 − 1 = 24, 36 − 1 = 35, 49 − 1 = 48, 64 − 1 = 63. The rule is n² − 1 for n = 2 to 8.

Only the 5th term breaks it: it should be 35, not 34. Differences confirm it: 5, 7, 9, 11, 13, 15 is the correct pattern, while 34 gives 10 and 14.

Answer: (d) 34

Common mistakes

Fitting a rule to the first 3 terms only: 2, 4, 8 could be × 2 or + 2, + 4; the 4th term (14 or 16) decides. Always test every term.

Missing an interleaved series: 4, 7, 8, 14, 16, 21 looks random until you split it into odd and even places.

Counting letters wrongly near the end of the alphabet: use 27 − n or the anchors T=20, Y=25 rather than counting on fingers.

In wrong-term questions, changing the term that looks odd without checking the others: after correcting, every term must fit.

Ignoring the second difference: when first differences are 4, 6, 8, the rule is not 'add 4'; the difference itself grows by 2.

Practice questions

Q1. 4, 7, 8, 14, 16, 21, 32, ? (a) 28 (b) 64 (c) 35 (d) 42

Q2. 2, 9, 28, 65, ? (a) 124 (b) 125 (c) 126 (d) 217

Q3. D, G, K, P, ? (a) U (b) V (c) W (d) T

Answers

Q1. (a) Odd places 4, 8, 16, 32 double; even places 7, 14, 21 add 7, so the next even-place term is 28.

Q2. (c) The terms are n³ + 1: 1 + 1, 8 + 1, 27 + 1, 64 + 1, so the next is 125 + 1 = 126.

Q3. (b) D(4), G(7), K(11), P(16): gaps +3, +4, +5, so next +6 gives 22 = V.

Quick revision

Steady growth: differences; doubling: × 2 ± c; near powers: n² ± k or n³ ± k; zig-zag: split odd and even places.

Write differences under the series; if needed, differences of differences.

Letters: positions, EJOTY, 27 − n; clusters: one series per position.

Wrong term: exactly one term must break the rule.

Concepts and rules

In a figure series, each figure is one step of a process. Every element changes by its own fixed rule at each step: it turns, moves along a path, changes in number, or alternates in shading.

Directions are numbered clockwise: N 0, NE 1, E 2, SE 3, S 4, SW 5, W 6, NW 7. A turn of 45° clockwise adds 1, 90° adds 2; anticlockwise subtracts. Work modulo 8, so 9 means 1 (NE).

Path along the border of a 3×3 grid, clockwise from the top-left: R1C1 (0), R1C2 (1), R1C3 (2), R2C3 (3), R3C3 (4), R3C2 (5), R3C1 (6), R2C1 (7). After 7 the path returns to 0.

Step patterns: constant (1, 1, 1), growing (1, 2, 3, 4), or alternating (1, 2, 1, 2). With growing steps the positions are 0, 1, 3, 6, 10, … taken modulo 8.

Counts follow number series: sides of polygons (3, 4, 5, 6), dots (1, 2, 4, 7, 11 with gaps +1, +2, +3, +4), or decreasing counts (5, 4, 3, 2).

Different elements can move in different directions and at different speeds in the same series; never assume they move together.

Method and shortcuts

Step 1: List the elements and write each one's state in every figure as a number (direction index, path index or count).

Step 2: For each element, write the differences between consecutive figures and name the step pattern.

Step 3: Extend each element one step, reduce direction and path indices modulo 8, and write the expected figure before looking at the options.

Shortcut: for alternation, compare figure 1 with figure 3; if they match in a feature, that feature repeats every 2 figures.

Shortcut: a move of 8 steps on the border or a turn of 360° brings an element back to its start, so only the remainder after dividing by 8 matters.

When it fails: if the steps look irregular, check whether they grow (1, 2, 3) or alternate (1, 2, 1, 2), and check the opposite direction of travel.

Example 1: Steady rotation

Question: An arrow points N, NE, E, SE in 4 figures. Where does it point in the 5th?

Solution: Indices 0, 1, 2, 3: the arrow turns 45° clockwise each time, so the next index is 4, which is S.

Answer: S

Example 2: Growing sides

Question: Triangle, Square, Pentagon, ?

Solution: Sides 3, 4, 5 increase by 1, so the next figure has 6 sides.

Answer: Hexagon

Example 3: Moving along the border

Question: A dot is at R1C1, R1C2, R1C3, R2C3 in 4 figures. Where is it in the 5th?

Solution: Path indices 0, 1, 2, 3: one step clockwise each time. Index 4 is R3C3.

Answer: R3C3

Example 4: Growing steps

Question: A dot is at R1C1, R1C2, R2C3, R3C1 in 4 figures. Where is it in the 5th? (a) R2C1 (b) R1C3 (c) R3C3 (d) R1C2

Solution: Path indices are 0, 1, 3, 6, so the steps are 1, 2, 3 clockwise. The next step is 4: 6 + 4 = 10, and 10 − 8 = 2, which is R1C3.

Option (a) assumes a step of 1 again, (c) a step of 2 from R3C1 the other way, and (d) forgets to wrap round.

Answer: (b) R1C3

Example 5: 2 elements moving differently

Question: Fig 1: arrow N, circle R1C1. Fig 2: arrow E, circle R2C1. Fig 3: arrow S, circle R3C1. Fig 4? (a) arrow W, circle R2C1 (b) arrow N, circle R3C2 (c) arrow W, circle R3C2 (d) arrow E, circle R3C3

Solution: Arrow indices 0, 2, 4: it turns 90° clockwise each time, so next is 6 = W.

Circle path indices 0, 7, 6: it moves 1 step anticlockwise each time, so next is 5 = R3C2. The arrow and the circle turn in opposite directions.

Answer: (c) arrow W, circle R3C2

Example 6: Growing turn and growing count

Question: 4 figures show an arrow and some dots: (N, 1 dot), (NE, 2 dots), (SE, 4 dots), (W, 7 dots). What is the 5th figure? (a) S, 11 dots (b) E, 10 dots (c) NW, 11 dots (d) E, 11 dots

Solution: Arrow indices 0, 1, 3, 6: turns of 1, 2, 3 steps (45°, 90°, 135°) clockwise. The next turn is 4 steps: 6 + 4 = 10, and 10 − 8 = 2 = E.

Dots 1, 2, 4, 7 have gaps +1, +2, +3, so the next count is 7 + 4 = 11. Option (b) gets the count wrong, (a) and (c) get the arrow wrong.

Answer: (d) E, 11 dots

Common mistakes

Assuming every element moves the same way: in Example 5 the arrow turns clockwise while the circle moves anticlockwise.

Forgetting to wrap round: index 10 on an 8-cell border is index 2, not an error.

Reading the grid path in the wrong order: write the 8 border cells in clockwise order once and use their numbers.

Checking only the direction and not the count, or the other way round; options are built to be right in one feature and wrong in another.

Practice questions

Q1. An arrow points N, W, S in 3 figures. Where does it point in the 4th? (a) N (b) E (c) W (d) NE

Q2. A dot is at R1C1, R1C3, R3C3 in 3 figures. Where is it in the 4th? (a) R3C1 (b) R2C1 (c) R1C1 (d) R3C2

Q3. Figures: triangle with 5 dots, square with 4 dots, pentagon with 3 dots, ? (a) hexagon with 3 dots (b) pentagon with 2 dots (c) heptagon with 2 dots (d) hexagon with 2 dots

Answers

Q1. (b) Indices 0, 6, 4: the arrow turns 90° anticlockwise each time, so next is 2 = E.

Q2. (a) Path indices 0, 2, 4: 2 steps clockwise each time, so next is 6 = R3C1.

Q3. (d) Sides 3, 4, 5 rise by 1 to 6 (hexagon); dots 5, 4, 3 fall by 1 to 2.

Quick revision

Directions: N 0, NE 1, E 2, SE 3, S 4, SW 5, W 6, NW 7; work modulo 8.

Border path: R1C1, R1C2, R1C3, R2C3, R3C3, R3C2, R3C1, R2C1 as 0 to 7.

Steps may be constant, growing or alternating; check each element separately.

Write the expected figure before reading the options.

Concepts and rules

In these questions you arrange words in a meaningful order and choose the option that lists their numbers in that order. For 1. Adult 2. Egg 3. Pupa 4. Larva the answer code 2, 4, 3, 1 means Egg, Larva, Pupa, Adult.

Life cycle: egg → larva → pupa → adult for butterflies and other insects with complete metamorphosis; eggs → tadpole → froglet → frog; seed → seedling → plant → flower → fruit.

Size or hierarchy: village → district → state → country → continent; letter → word → phrase → sentence → paragraph → chapter → book.

Process and time: ploughing → sowing → irrigation → harvesting → selling; illness → consultation → diagnosis → treatment → recovery. Ask what must happen before each step.

Dictionary order: compare words letter by letter from the left; the first letter where they differ decides. Path comes before Patrol because h comes before r at the 4th letter.

If one word is exactly the beginning of another (Car, Card), the shorter word comes first. Word length matters only in this case.

Method and shortcuts

Step 1: Name the theme: life cycle, size, process, time, or dictionary order.

Step 2: Fix the clear first and last items (the smallest and largest, the start and end).

Step 3: Place the middle items by asking 'what must come just before this?'

Step 4: Eliminate options using the first item only, then compare the survivors where they differ; often 2 options differ by one swap in the middle.

Dictionary shortcut: write the common beginning once (for example 'Cont'), then list only the next letters of each word and sort those.

When it fails: if 2 orders seem possible, look for the step that physically cannot happen earlier (you cannot harvest before sowing; diagnosis needs a consultation).

Example 1: Life cycle

Question: Arrange in a logical order: 1. Adult 2. Egg 3. Pupa 4. Larva

Solution: An insect such as a butterfly starts as an egg, hatches into a larva, becomes a pupa and emerges as an adult.

Answer: 2, 4, 3, 1

Example 2: Size

Question: Arrange from smallest to largest: 1. State 2. Village 3. Country 4. District

Solution: Villages make up a district, districts make up a state, and states make up the country.

Answer: 2, 4, 1, 3

Example 3: Dictionary order

Question: Arrange in dictionary order: 1. Patron 2. Pattern 3. Patrol 4. Path

Solution: All begin with 'Pat'. The 4th letters are r (Patron), t (Pattern), r (Patrol), h (Path), so Path comes first and Pattern last. Patrol and Patron share 'Patro'; at the 6th letter l comes before n.

Order: Path, Patrol, Patron, Pattern.

Answer: 4, 3, 1, 2

Example 4: Farming process

Question: Arrange in a logical order: 1. Harvesting 2. Sowing 3. Ploughing 4. Irrigation 5. Selling (a) 3, 2, 4, 1, 5 (b) 2, 3, 4, 1, 5 (c) 3, 2, 1, 4, 5 (d) 3, 4, 2, 1, 5

Solution: The field is ploughed first, then seeds are sown, the growing crop is irrigated, the crop is harvested, and finally the produce is sold.

Option (b) sows before ploughing, (c) harvests before irrigating, and (d) irrigates before sowing.

Answer: (a) 3, 2, 4, 1, 5

Example 5: Dictionary order with a long common start

Question: Arrange in dictionary order: 1. Contrast 2. Contract 3. Contour 4. Control 5. Contrary (a) 3, 2, 1, 5, 4 (b) 2, 5, 1, 4, 3 (c) 3, 2, 5, 1, 4 (d) 3, 5, 2, 1, 4

Solution: All begin with 'Cont'. The 5th letter is o in Contour and r in the rest, so Contour comes first.

Among 'Contr' words the 6th letter is a in Contract, Contrary, Contrast and o in Control, so Control comes last. Among 'Contra' words the 7th letters are c, r, s: Contract, Contrary, Contrast.

Order: Contour, Contract, Contrary, Contrast, Control.

Answer: (c) 3, 2, 5, 1, 4

Example 6: Sequence of events

Question: Arrange in a logical order: 1. Recovery 2. Diagnosis 3. Illness 4. Treatment 5. Consultation (a) 3, 2, 5, 4, 1 (b) 5, 3, 2, 4, 1 (c) 3, 5, 4, 2, 1 (d) 3, 5, 2, 4, 1

Solution: A person falls ill, consults a doctor, the doctor diagnoses the illness, treatment follows the diagnosis, and recovery comes last.

Option (a) diagnoses before the consultation, (b) consults before falling ill, and (c) treats before diagnosing.

Answer: (d) 3, 5, 2, 4, 1

Common mistakes

Sorting dictionary words by length: length matters only when one word is the beginning of another (Car before Card). Otherwise the first differing letter decides: Baby comes before Bat because b comes before t at the 3rd letter.

Stopping at the first common letters: in Patrol and Patron the first 5 letters match; go on to the 6th letter.

Mixing up larva and pupa: the larva (caterpillar) feeds and grows; the pupa is the resting stage before the adult.

Reading the answer code backwards: the code lists the word numbers in order of position, not the position of each word.

Practice questions

Q1. Arrange from smallest to largest: 1. Sentence 2. Letter 3. Word 4. Paragraph 5. Phrase (a) 2, 3, 1, 5, 4 (b) 2, 3, 5, 1, 4 (c) 3, 2, 5, 1, 4 (d) 2, 5, 3, 1, 4

Q2. Arrange in dictionary order: 1. Seclude 2. Secure 3. Section 4. Secret 5. Second (a) 5, 1, 4, 3, 2 (b) 1, 5, 3, 4, 2 (c) 1, 4, 5, 3, 2 (d) 1, 5, 4, 3, 2

Q3. Arrange in a logical order: 1. Frog 2. Tadpole 3. Eggs 4. Froglet (a) 3, 4, 2, 1 (b) 2, 3, 4, 1 (c) 3, 2, 4, 1 (d) 3, 2, 1, 4

Answers

Q1. (b) Letter, Word, Phrase, Sentence, Paragraph: each is built from the one before.

Q2. (d) After 'Sec' the next letters are l (Seclude), o (Second), r (Secret), t (Section), u (Secure).

Q3. (c) Eggs hatch into tadpoles, tadpoles grow legs and become froglets, and froglets grow into frogs.

Quick revision

Name the theme, then fix the first and last items.

Ask 'what must come just before this?' for the middle items.

Dictionary order: first differing letter decides; a word that begins a longer word comes first.

Eliminate options with the first item, then check the middle swap.