Lessons / Logical reasoning
Logical reasoning · Day 9

Series

Every series is built from one of a handful of rules. Check them in order: differences, second differences, ratios, squares/cubes, two series interleaved.

Practise (12 questions) Present as slidesCheat sheet: Summations and series

Why this topic matters

2, 6, 12, 20, 30, ? Write the gaps underneath: 4, 6, 8, 10. The next gap is 12, so the answer is 42. Most series give up their secret in the gaps.

Number and letter series are among the fastest marks in reasoning, if you have a fixed checklist. Without one, students stare at the numbers.

The checklist for number series

1. Differences: write the gaps. If the gaps are not constant, write the gaps of the gaps.

2. Ratios: if numbers grow fast, divide each by the one before (x 2, x 3, or × 2 + 1 ...).

3. Squares and cubes: look for n2, n3, n2 + 1, n3 - 1.

4. Two series mixed: odd positions follow one rule, even positions another.

5. Primes: 2, 3, 5, 7, 11, 13 ... hidden in the numbers or the gaps.

26122030+4+6+8+10+2+2+2
Gaps 4, 6, 8, 10 grow by 2. Next gap 12, next term 42
Example 13, 7, 15, 31, 63, ?
  1. Gaps: 4, 8, 16, 32. They double.Step 1 of the checklist.
  2. Also each term = previous × 2 + 1.Two ways to see it; both agree.
  3. 63 × 2 + 1 = 127.
Example 21, 8, 27, 64, ?, 216
  1. These are 13, 23, 33, 43.Step 3: cubes.
  2. 53 = 125.
Example 34, 5, 12, 10, 36, 15, ?
  1. Odd places: 4, 12, 36: x 3 each time.Step 4: two series woven together.
  2. Even places: 5, 10, 15: + 5.
  3. Next is odd place: 36 × 3 = 108.

Wrong number in a series

Find the rule from most of the terms, then check each term. The one that breaks the rule is wrong. Check the terms around it too: sometimes fixing one term fixes two gaps.

Example 1Find the wrong number: 2, 5, 10, 17, 26, 38, 50
  1. Gaps: 3, 5, 7, 9, 12, 12.
  2. Should be 3, 5, 7, 9, 11, 13 (odd numbers).Most gaps follow odd numbers.
  3. 26 + 11 = 37, so 38 is wrong (and 37 + 13 = 50 confirms it).

Letter series

Turn letters into numbers (A = 1 ... Z = 26) and treat it as a number series. Write the alphabet with positions at the top of your sheet.

Remember EJOTY: E = 5, J = 10, O = 15, T = 20, Y = 25. They let you find any letter's position in two steps.

A126B225C324D423E522F621G720H819I918J1017K1116L1215M1314N1413O1512P1611Q1710R189S198T207U216V225W234X243Y252Z261
Letter positions: A=1 ... Z=26 (reverse position underneath). Opposite pairs add to 27
Example 1B, E, I, N, ?
  1. Positions 2, 5, 9, 14.Letters to numbers.
  2. Gaps 3, 4, 5, so next gap 6.
  3. 14 + 6 = 20 = T.
Example 2AZ, BY, CX, ?
  1. First letters A, B, C go forward. Second letters Z, Y, X go backward.Each letter pair is an opposite pair (adds to 27).
  2. DW.

Formula summary

Differences
Write the gaps. Constant → AP. Gaps in AP → second difference constant.
Ratios
x2, x3, or x2 +1 style (multiply then add).
Squares / cubes
n2 +/- 1, n3 +/- n. Memorise squares to 30 and cubes to 15.
Alternate series
Odd positions and even positions follow two different rules.
Letter series
Convert to positions (A = 1 ... Z = 26). EJOTY = 5, 10, 15, 20, 25.
Wrong number
Find the rule from the majority, then the odd term out.

Shortcuts and tricks

Letter positions by anchors

Use it when: Converting letters fast.

  1. EJOTY: E5 J10 O15 T20 Y25.
  2. Reverse position = 27 - position (A <→ Z, M <→ N).
ExampleNext: B, E, J, Q, ?
  1. 2, 5, 10, 17 = n2 + 1.
  2. Next 26 → Z.

Before you move on, you should be able to...

  • run the five-step checklist
  • write gaps and gaps of gaps
  • convert letters to positions with EJOTY

Practice questions (12)

Try each one before opening the solution. Or practise them one by one so your score is saved.

easy 2, 6, 12, 20, 30, ?
  1. n(n + 1).

Answer: 42

easy 3, 7, 15, 31, 63, ?
  1. x2 + 1.

Answer: 127

easy 1, 8, 27, 64, 125, ?
  1. cubes.

Answer: 216

easy 5, 11, 23, 47, 95, ?
  1. x2 + 1.

Answer: 191

medium Wrong number: 2, 5, 10, 17, 26, 38, 50
  1. n2 + 1: 2, 5, 10, 17, 26, 37, 50.
  2. 38 should be 37.

Answer: 38

medium 4, 6, 12, 14, 28, 30, ?
  1. +2, x2 alternately.

Answer: 60

medium AZ, CX, EV, GT, ?
  1. First letter +2, second -2.

Answer: IR

hard 1, 2, 6, 15, 31, 56, ?
  1. Gaps 1, 4, 9, 16, 25 = squares.
  2. Next gap 36.

Answer: 92

hard 3, 5, 9, 17, 33, ?
  1. x2 - 1.

Answer: 65

hard 7, 26, 63, 124, ?
  1. n3 - 1 for n = 2..6.

Answer: 215

hard 1, 4, 27, 256, ?
  1. nn.

Answer: 3125

hard 2, 3, 5, 7, 11, 13, ? (next two terms)
  1. Primes.

Answer: 17, 19

Class plan (2 hours, for trainers)

A tested order for teaching this topic in one 2-hour session. Present mode follows the same order.

10 minWarm-up: squares and cubes quiz.
25 minThe five-step checklist with the gap ladder.
15 minTwo woven series.
15 minWrong number.
20 minLetter series, EJOTY, opposite letters.
30 minPractice set, timed.
5 minRecap.
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