38  Series and Sequences

38.1 Introduction

Series and sequences test your ability to identify patterns, rules, and progressions in numbers, letters, or figures. These questions are common in reasoning and aptitude exams. Mastery requires recognizing arithmetic, geometric, and mixed progressions, as well as spotting anomalies and predicting the next/following term.

38.2 1) Types of Numerical Sequences

38.2.1 1.1 Arithmetic Progression (AP)

  • Difference between consecutive terms is constant.
  • General form: \(a, a+d, a+2d, a+3d, \dots\)
  • \(n\)th term: \(T_n = a + (n-1)d\)
  • Sum of first \(n\) terms: \(S_n = \frac{n}{2}[2a + (n-1)d]\)

Example: 3, 6, 9, 12, … (common difference \(d=3\)).

38.2.2 1.2 Geometric Progression (GP)

  • Ratio between consecutive terms is constant.
  • General form: \(a, ar, ar^2, ar^3, \dots\)
  • \(n\)th term: \(T_n = ar^{n-1}\)
  • Sum of first \(n\) terms: \(S_n = \frac{a(r^n-1)}{r-1}, \; r\neq1\)

Example: 2, 4, 8, 16, … (common ratio \(r=2\)).

38.2.3 1.3 Harmonic Progression (HP)

  • Reciprocals of the terms form an AP.
  • Example: 1, 1/2, 1/3, 1/4, …

38.2.4 1.4 Mixed Progressions

Combination of AP and GP, or patterns with squares, cubes, prime numbers, Fibonacci series, etc.

Examples:
- 1, 4, 9, 16, 25, … (perfect squares).
- 2, 3, 5, 7, 11, … (prime numbers).
- 1, 1, 2, 3, 5, 8, … (Fibonacci: \(T_n = T_{n-1}+T_{n-2}\)).

38.3 2) Types of Reasoning Series

38.3.1 2.1 Missing Term Series

Identify the missing number/letter/figure based on the given pattern.
Example: 2, 4, 8, ?, 32 → Rule: ×2 → Answer = 16.

38.3.2 2.2 Wrong Term Series

Find the odd element that doesn’t follow the pattern.
Example: 3, 6, 12, 24, 50, 96 → Rule: ×2 → Wrong term = 50.

38.3.3 2.3 Alphabetic/Alphanumeric Series

  • Pure alphabetic: A, C, E, G, … (skip one letter).
  • Alphanumeric: A1, B2, C3, D4, …
  • Complex patterns may mix arithmetic shifts with positions.

38.3.4 2.4 Visual/Non-Verbal Series

Series with shapes, rotations, mirror images, or shaded patterns.

38.4 3) Problem-Solving Strategy

  1. Look for common difference (AP).
  2. Check common ratio (GP).
  3. Test squares, cubes, primes, Fibonacci.
  4. For letters: map to ASCII or alphabet positions (A=1, B=2, …).
  5. For figures: check symmetry, rotation, shading.

38.5 4) Solved Examples

38.5.1 Example 1

Find the missing term: 7, 14, 28, ?, 112.
- Rule: ×2 each step.
- Missing term = 56.

38.5.2 Example 2

Identify the wrong term: 4, 8, 16, 32, 64, 130, 256.
- Pattern: powers of 2 → Wrong = 130.

38.5.3 Example 3

Fill in the blank: A, D, G, J, ?
- Rule: +3 in alphabet positions.
- Answer = M.

38.5.4 Example 4

Series: 2, 6, 12, 20, 30, ?
- Differences = 4, 6, 8, 10, … → Next difference = 12.
- Missing term = 42.

38.6 5) Practice Questions

  1. 5, 10, 20, 40, ?
  2. 3, 6, 18, 72, ?
  3. 121, 144, 169, 196, ?
  4. 2, 5, 10, 17, 26, ?
  5. A, Z, B, Y, C, X, ?
  6. 8, 27, 64, 125, ?
  7. 1, 4, 9, 16, 23, ? (find wrong term)
  8. Fibonacci series: 1, 1, 2, 3, 5, 8, ?

38.7 6) Answer Key

  1. 80 (×2 each step).
  2. 216 (×3, ×3, ×4, …).
  3. 225 (next square = 15²).
  4. 37 (differences = +3, +5, +7, +9, +11).
  5. D (pattern: A↔︎Z, B↔︎Y, C↔︎X, D↔︎W).
  6. 216 (cubes: 2³, 3³, 4³, 5³, 6³).
  7. Wrong term = 23 (should be 25).
  8. 13 (Fibonacci rule).

38.8 Summary

  • Recognize AP, GP, HP, and mixed patterns.
  • Use differences, ratios, or special number families (squares, cubes, primes).
  • For alphabets, use positional logic.
  • For non-verbal, focus on visual transformations.
    With practice, series problems become quick mental exercises.