Series and Analogies
Number series, letter series, analogies and odd-one-out for CGPSC CSAT: difference tables, EJOTY, opposite letters and a 60-second speed drill.
Prerequisites
- Squares up to 25 and cubes up to 12 — at least half of all number series are built on them.
- Multiplication tables up to 20 and quick subtraction (the difference method below is pure subtraction).
- This is the first topic of General Mental Ability. The number sense from Basic Numeracy helps throughout.
The Hook: The Paddy Stack at the Procurement Centre
At a paddy procurement centre (धान खरीदी केंद्र) near Dhamtari, a worker stacks gunny bags in a triangle: 1 bag on the top row, then 2, then 3, and so on. The running totals the clerk writes down are 1, 3, 6, 10, 15, …
What comes next? The gaps between the totals are 2, 3, 4, 5 — so the next gap is 6, and the next total is 21.
The question this page answers: how do you spot the hidden rule in a number or letter sequence, or in a pair of related words, within a few seconds — and check it before you mark the answer?
Why it matters for CGPSC: series (श्रेणी) and analogies (सादृश्यता) are the most predictable part of the “general mental ability” (सामान्य मानसिक योग्यता) block of Prelims Paper II (CSAT). The patterns repeat from paper to paper, so practice converts almost directly into marks. The reasoning part of Mains Paper IV (GS-II) uses the same patterns.
Number Series: The Five-Step Check
A number series follows one rule. Test the common rules in this order — it is roughly the order of how often they appear:
| Step | Test | Signal | Example |
|---|---|---|---|
| 1 | Differences | Terms grow slowly and steadily | 3, 7, 13, 21, 31 → gaps 4, 6, 8, 10 → next 43 |
| 2 | Ratios | Terms roughly double or triple | 5, 11, 23, 47, 95 → each \( \times 2 + 1 \) → next 191 |
| 3 | Squares / cubes | Terms near 1, 4, 9, 16 … or 1, 8, 27 … | 2, 6, 12, 20, 30 = \( n(n+1) \) → next 42 |
| 4 | Primes | Odd terms, irregular small gaps | 2, 3, 5, 7, 11, 13 → next 17 |
| 5 | Alternating series | Terms jump up and down | 2, 20, 4, 17, 6, 14, 8 → two series → next 11 |
Step 1: The difference table
Write the gaps under the series. If the first gaps are not constant, take the gaps of the gaps (second differences).
4, 6, 10, 18, 34, ?
First differences: 2, 4, 8, 16 — they double. The next difference is 32, so the answer is \( 34 + 32 = 66 \).
Step 2: Multiply-and-add patterns
When terms roughly double or triple, test \( \times k \pm c \). A favourite variant changes the multiplier each step.
3, 4, 10, 33, 136, ?
\( 3\times 1 + 1 = 4 \), \( 4\times 2 + 2 = 10 \), \( 10\times 3 + 3 = 33 \), \( 33\times 4 + 4 = 136 \). Next: \( 136\times 5 + 5 = 685 \).
Another common one is the factorial series: 1, 2, 6, 24, 120, ? — multiply by 2, 3, 4, 5, then 6 → 720.
Step 3: Squares, cubes and their neighbours
| Pattern | Series | Next |
|---|---|---|
| \( n^2 \) | 1, 4, 9, 16, 25 | 36 |
| \( n^3 \) | 1, 8, 27, 64 | 125 |
| \( n^2 + 1 \) | 2, 5, 10, 17, 26 | 37 |
| \( n^2 - 1 \) | 0, 3, 8, 15, 24 | 35 |
| \( n(n+1) \) | 2, 6, 12, 20, 30 | 42 |
| \( n^3 + 1 \) | 2, 9, 28, 65 | 126 |
| Triangular \( \frac{n(n+1)}{2} \) | 1, 3, 6, 10, 15 | 21 |
Step 4 and 5: Primes and two series in one
If the gaps look random but all terms are small and odd (except 2), check the prime list: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47.
If a series jumps up and down, split it into odd-position and even-position terms.
2, 20, 4, 17, 6, 14, 8, ?
Odd positions: 2, 4, 6, 8 (+2). Even positions: 20, 17, 14 (−3). The missing term is in an even position, so it is \( 14 - 3 = 11 \).
Wrong-term questions
“Find the wrong term” is a number series in disguise: find the rule from the majority of terms, then locate the one term that breaks it.
Letter Series
Every letter question is a number question once you convert letters to their positions.
| 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 |
Opposite letters (each pair adds to 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.
- B, E, H, K, ? → 2, 5, 8, 11 (+3) → 14 = N.
- A, C, F, J, O, ? → 1, 3, 6, 10, 15 (gaps +2, +3, +4, +5) → gap +6 → 21 = U.
- AZ, BY, CX, ? → the first letter moves forward, the second backward; each pair is an opposite pair → DW.
Analogies
An analogy (सादृश्यता) gives a pair with a relation and asks you to complete a second pair with the same relation: A : B :: C : ?
Number analogies
| Pair | Relation | Apply to |
|---|---|---|
| 3 : 27 | \( n^3 \) | 5 : 125 |
| 4 : 17 | \( n^2 + 1 \) | 6 : 37 |
| 7 : 56 | \( n(n+1) \) | 9 : 90 |
Always check that your rule fits the given pair exactly; two rules can fit one number but rarely both.
Letter analogies
ACE : BDF :: GIK : ? — every letter moves forward by one → HJL.
Word analogies
Name the relation in a short sentence ("___ is the capital of ___"), then test each option with the same sentence.
| Relation | Example |
|---|---|
| Capital – state | Raipur : Chhattisgarh :: Bhopal : Madhya Pradesh |
| Dam – river | Hirakud : Mahanadi :: Bhakra : Sutlej |
| Worker – workplace | Teacher : School :: Doctor : Hospital |
| Tool – function | Pen : Write :: Knife : Cut |
| Part – whole | Petal : Flower :: Page : Book |
| Young one | Cow : Calf :: Dog : Puppy |
| Unit – quantity | Ampere : Current :: Volt : Potential difference |
Odd One Out (Classification)
Classification asks which item does not share the property of the rest. Find the property from the majority first.
- 9, 25, 49, 81, 121, 143 → all others are perfect squares; \( 143 = 11\times 13 \) is the odd one.
- 31, 37, 41, 49, 53 → all others are prime; \( 49 = 7^2 \).
- ACE, GIK, MOQ, RTW → the first three skip one letter each time; RTW skips one, then two — RTW is odd.
Memory Tricks
- EJOTY = 5, 10, 15, 20, 25. From E, J, O, T, Y you can reach any letter in one or two steps: R = T − 2 = 18.
- Opposite pairs add to 27. Easy anchors: IR (Indian Railways), HS (High School), GT (GT Road), MN (the middle of the alphabet).
- The five-step check order: “D-R-S-P-A” — Differences, Ratios, Squares/cubes, Primes, Alternating.
- Two-sentence rule for analogies: say the relation as a sentence, then say it again with each option.
Don’t Confuse
| This | vs | That |
|---|---|---|
| \( n(n+1) \): 2, 6, 12, 20 | \( \frac{n(n+1)}{2} \): 1, 3, 6, 10 | |
| \( n^2 + 1 \): 2, 5, 10, 17 | \( n^2 - 1 \): 0, 3, 8, 15 | |
| Position from start (R = 18) | Position from end (R = 27 − 18 = 9) | |
| Missing term in an alternating series — check its position | Next term in a simple series — just continue the rule |
Common Mistakes
Practice
Level 1: Foundation
- What is the position of the letter T in the alphabet? Of its opposite letter?
- Next term: 1, 8, 27, 64, ?
- Next term: 2, 3, 5, 7, 11, 13, ?
- Complete: Raipur : Chhattisgarh :: Bhopal : ?
- Odd one out: 31, 37, 41, 49, 53
Answers
- T = 20; its opposite is \( 27 - 20 = 7 \) = G.
- 125 (cubes).
- 17 (primes).
- Madhya Pradesh (capital – state).
- 49 — the only non-prime.
Level 2: Prelims MCQs
Q1. Find the next term: 4, 6, 10, 18, 34, ?
(a) 50 (b) 64 (c) 66 (d) 68
Answer: (c) 66
Q2. Find the wrong term in the series: 1, 4, 9, 16, 24, 36, 49
(a) 4 (b) 16 (c) 36 (d) 24
Answer: (d) 24
Q3. Find the next letter: A, C, F, J, O, ?
(a) T (b) U (c) V (d) S
Answer: (b) U
Q4. Match List I (series) with List II (next term):
| List I | List II |
|---|---|
| A. 1, 8, 27, 64, ? | 1. 13 |
| B. 2, 3, 5, 7, 11, ? | 2. 31 |
| C. 1, 2, 6, 24, ? | 3. 120 |
| D. 3, 7, 13, 21, ? | 4. 125 |
(a) A-4, B-1, C-3, D-2 (b) A-4, B-2, C-3, D-1 (c) A-3, B-1, C-4, D-2 (d) A-1, B-4, C-2, D-3
Answer: (a) A-4, B-1, C-3, D-2
Q5. Consider the following statements about the series 2, 6, 12, 20, 30, …:
- The next term is 42.
- The 10th term is 110.
- The difference between consecutive terms is constant.
Which of the statements given above is/are correct?
(a) 1 only (b) 1 and 2 only (c) 2 and 3 only (d) 1, 2 and 3
Answer: (b) 1 and 2 only
Level 3: Speed Drill — Solve in 60 Seconds
The reasoning part of Mains Paper IV (GS-II) and Prelims Paper II both reward a fixed routine. Each item below should take under 60 seconds using the five-step check, EJOTY or the 27 rule.
- 3, 4, 10, 33, 136, ?
- 2, 20, 4, 17, 6, 14, 8, ?
- B, E, H, K, ?
- ACE : BDF :: GIK : ?
- Odd one out: 9, 25, 49, 81, 121, 143
- AZ, BY, CX, ?
- 3 : 27 :: 5 : ?
Answers with the method used
- 685 — \( \times 1 + 1, \times 2 + 2, \times 3 + 3 \dots \); \( 136\times 5 + 5 \).
- 11 — even-position series 20, 17, 14 (−3).
- N — positions 2, 5, 8, 11 (+3) → 14.
- HJL — each letter +1.
- 143 — \( 11\times 13 \), not a square.
- DW — opposite pairs, first letter moving forward.
- 125 — cube.
Quick Revision Table
| Idea | Fact |
|---|---|
| Check order | Differences → Ratios → Squares/cubes → Primes → Alternating |
| Second differences | Use when first differences are not constant |
| \( n(n+1) \) | 2, 6, 12, 20, 30, 42 |
| Triangular numbers | 1, 3, 6, 10, 15, 21 |
| EJOTY | E = 5, J = 10, O = 15, T = 20, Y = 25 |
| Opposite letter | \( 27 - \) position |
| Analogy | State the relation as a sentence and test every option |
| Odd one out | Find the majority property first |
Key Takeaways
- Most number series fall to the difference table; if not, test ratios, squares/cubes, primes and alternating series in that order.
- Convert letters to numbers with EJOTY and the 27 rule — never count on fingers.
- An analogy is a sentence: name the relation, then test each option with the same sentence.
- In alternating series, the missing term belongs to the sub-series that owns its position.
- Always verify your rule on every given term before marking — negative marking punishes half-checked rules.
What’s Next
- Puzzles — ranking, ordering, grid and counting puzzles.
- Calendars and Clocks — the two fixed-formula topics of mental ability.
- Logical Reasoning — coding-decoding uses the same letter-position skills.
- Basic Numeracy — squares, cubes and quick arithmetic behind every number series.
- General Mental Ability — Quick Revision — the whole chapter on one page.