Number Series and Coding-Decoding: Patterns, Method and Examples

Series and coding-decoding questions are a regular part of the mental ability (IQ) section of Lok Sewa exams. They do not need long calculations; they need a calm, organised way of spotting the rule. This note shows the patterns that come up again and again, gives an alphabet table to learn by heart, and explains a step-by-step method with worked examples.

1. What a series question asks

A series is a list of numbers or letters arranged by a fixed rule. You are asked to find the next term, a missing term, or the one term that does not fit. A good answer has one clear rule that works for every term, not just the first two or three.

Step-by-step method for series:

• Look at the size of the numbers. Slow, steady growth suggests adding; fast growth suggests multiplying or powers.
• Write the differences between neighbouring terms under the series.
• If the differences have no pattern, try ratios (divide each term by the one before).
• Check for squares, cubes or prime numbers.
• If nothing works, split the series into two alternating series (odd places and even places).
• Test your rule on every term before choosing an option.

2. Add and subtract patterns

Constant difference: the same number is added or subtracted each time.

• 3, 7, 11, 15, ? Differences are all +4, so the answer is 19.
• 50, 44, 38, 32, ? Differences are all −6, so the answer is 26.

Changing difference: the differences themselves form a pattern.

• 2, 5, 10, 17, 26, ? Differences are 3, 5, 7, 9, so the next difference is 11 and the answer is 37.
• 1, 2, 4, 7, 11, ? Differences are 1, 2, 3, 4, so the next is 5 and the answer is 16.

3. Multiply and divide patterns

• 4, 12, 36, 108, ? Each term is × 3, so the answer is 324.
• 256, 128, 64, 32, ? Each term is ÷ 2, so the answer is 16.
• 1, 2, 6, 24, 120, ? The multipliers grow: × 2, × 3, × 4, × 5, so the next is × 6 and the answer is 720.

4. Squares, cubes and prime numbers

Learn these lists so that you recognise them at once:

• 4, 9, 16, 25, ? These are 2², 3², 4², 5², so the answer is 6² = 36.
• 1, 8, 27, 64, ? These are cubes, so the answer is 5³ = 125.
• 0, 3, 8, 15, 24, ? These are one less than squares (1 − 1, 4 − 1, 9 − 1 …), so the answer is 36 − 1 = 35.

Prime numbers have exactly two factors, 1 and themselves. The primes up to 50 are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43 and 47. Note that 1 is not a prime and 2 is the only even prime.

• 2, 3, 5, 7, 11, 13, ? The next prime is 17.

5. Alternating and mixed series

Alternating series: two different series are mixed in turn.

• 2, 10, 4, 20, 6, 30, ? Odd places: 2, 4, 6 (add 2). Even places: 10, 20, 30 (add 10). The next term is in an odd place, so the answer is 8.
• 5, 10, 8, 16, 14, ? The operations alternate: × 2, − 2, × 2, − 2. The next step is × 2, so the answer is 14 × 2 = 28.

Mixed operation: each term is made by two operations together.

• 5, 11, 23, 47, ? Rule: × 2 then + 1. Answer: 47 × 2 + 1 = 95.

Wrong term: find the one term that breaks the rule.

• 3, 5, 9, 17, 33, 66. Rule: × 2 then − 1, which gives 3, 5, 9, 17, 33, 65. So 66 is the wrong term.

6. Letter series and alphabet positions

Letter series use the position of each letter in the alphabet. The reverse position counts from Z backwards. A letter’s position plus its reverse position always equals 27.

Memory aid: the word EJOTY gives the letters at positions 5, 10, 15, 20 and 25. From there you can count forward or back a few steps.

Worked examples:

• B, E, H, K, ? Positions 2, 5, 8, 11 (add 3). Next is 14, which is N.
• Z, X, V, T, ? Each step goes back 2. Next is R.
• AZ, BY, CX, ? The first letter moves forward and the second moves back; each pair are opposite letters. Next is DW.

7. Coding-decoding: letter shift

In coding-decoding, a word is changed into a code by a fixed rule. You find the rule from the example and apply it to the new word.

Letter shift: each letter moves forward or backward by a fixed number of places.

• If CAT is written as DBU, each letter moves +1. So DOG becomes EPH.
• If BOOK is coded as CPPL, then DOOR is coded as EPPS (+1 each).
• If FISH is coded as DGQF, each letter moves −2 (F to D, I to G, S to Q, H to F).

Sometimes the shift changes by position, for example +1, +2, +3, +4 for the first, second, third and fourth letters. Always check every letter, not just the first one.

8. Coding-decoding: reverse and opposite letters

• Reverse order: the letters are written backwards. PEN becomes NEP, and LAMP becomes PMAL.
• Opposite letters: each letter is replaced by its partner from the other end of the alphabet (A with Z, B with Y, and so on). Since position plus reverse position is 27, find the partner by taking 27 minus the position. GOOD becomes TLLW (G 7 to T 20, O 15 to L 12, D 4 to W 23).
• Reverse plus shift: some codes first reverse the word and then shift the letters. Test both steps against the example.

9. Number coding and substitution

• Position coding: each letter is written as its position number. BAD becomes 2 1 4.
• Sum coding: a word is coded as the total of its letter positions. If CAT = 24 (3 + 1 + 20), then DOG = 4 + 15 + 7 = 26.
• Substitution: names of objects are swapped. If “pen” is called “book”, “book” is called “bag” and “bag” is called “box”, then the thing we carry books in (a bag) is called box. Always answer with the new name, not the real one.