Series Exit
Logical reasoning

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.

Start with this

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

Worked example

3, 7, 15, 31, 63, ?

  1. Gaps: 4, 8, 16, 32. They double.
    Why: Step 1 of the checklist.
  2. Also each term = previous × 2 + 1.
    Why: Two ways to see it; both agree.
  3. 63 × 2 + 1 = 127.

Answer: 127

Worked example

1, 8, 27, 64, ?, 216

  1. These are 13, 23, 33, 43.
    Why: Step 3: cubes.
  2. 53 = 125.

Answer: 125

Worked example

4, 5, 12, 10, 36, 15, ?

  1. Odd places: 4, 12, 36: x 3 each time.
    Why: Step 4: two series woven together.
  2. Even places: 5, 10, 15: + 5.
  3. Next is odd place: 36 × 3 = 108.

Answer: 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.

Worked example

Find 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).
    Why: Most gaps follow odd numbers.
  3. 26 + 11 = 37, so 38 is wrong (and 37 + 13 = 50 confirms it).

Answer: 38

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

Worked example

B, E, I, N, ?

  1. Positions 2, 5, 9, 14.
    Why: Letters to numbers.
  2. Gaps 3, 4, 5, so next gap 6.
  3. 14 + 6 = 20 = T.

Answer: T

Worked example

AZ, BY, CX, ?

  1. First letters A, B, C go forward. Second letters Z, Y, X go backward.
    Why: Each letter pair is an opposite pair (adds to 27).
  2. DW.

Answer: DW

Shortcut: 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).

Letter positions by anchors: try it

Next: B, E, J, Q, ?

  1. 2, 5, 10, 17 = n2 + 1.
  2. Next 26 → Z.

Answer: Z

Your turn 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

Your turn medium

4, 6, 12, 14, 28, 30, ?

  1. +2, x2 alternately.

Answer: 60

Your turn medium

AZ, CX, EV, GT, ?

  1. First letter +2, second -2.

Answer: IR

Your turn hard

1, 2, 6, 15, 31, 56, ?

  1. Gaps 1, 4, 9, 16, 25 = squares.
  2. Next gap 36.

Answer: 92

Recap

Now practise

12 questions with full solutions.

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