Series • Number, letter, and figure series

 

Series: Full-Length Notes by a Senior Professor

(Number, Letter, and Figure Series—With Examples, Explanations, and 100 Practice Questions)

Introduction to Series

A series is an ordered sequence of numbers, letters, or figures that follow a specific pattern or rule. Series questions are a staple in reasoning, aptitude, and competitive exams because they test your ability to recognize patterns, logical progression, and analytical thinking.

Types of Series

1. Number Series

Number series involve a sequence of numbers following a particular mathematical pattern. Your task is to identify the pattern and find the missing term(s).

Common Patterns in Number Series

  • Arithmetic Series: Each term is obtained by adding or subtracting a fixed number (common difference) to the previous term.
    Example: 4, 9, 14, 19, 24, ... (Add 5 each time)467

  • Geometric Series: Each term is obtained by multiplying or dividing the previous term by a fixed number (common ratio).
    Example: 3, 6, 12, 24, 48, ... (Multiply by 2)147

  • Square Number Series: Each term is a perfect square.
    Example: 1, 4, 9, 16, 25, ... (1², 2², 3², ...)146

  • Cube Number Series: Each term is a perfect cube.
    Example: 1, 8, 27, 64, 125, ... (1³, 2³, 3³, ...)146

  • Fibonacci Series: Each term is the sum of the two preceding terms.
    Example: 1, 1, 2, 3, 5, 8, 13, ...3

  • Triangular Number Series: Each term represents the number of dots that can form an equilateral triangle.
    Example: 1, 3, 6, 10, 15, 21, ... (Add 2, then 3, then 4, ...)15

  • Two-Stage Series: The difference between terms forms another arithmetic series.
    Example: 1, 3, 6, 10, 15, ... (Differences: 2, 3, 4, 5)56

  • Mixed Series: A combination of operations or changing patterns.
    Example: 7, 16, 34, 70, 142, ... (Multiply by 2, add 2)5

Tips for Solving Number Series

  • Check for simple addition/subtraction or multiplication/division.

  • Look for squares, cubes, or other mathematical operations.

  • Analyze the difference or ratio between terms.

  • For complex series, look for a pattern in the differences or ratios themselves.

2. Letter Series

Letter series involve sequences of alphabets that follow a logical pattern. These patterns may be based on the position of the letters in the alphabet, skipping letters, or alternating sequences.

Common Patterns in Letter Series

  • Consecutive Letters:
    Example: A, B, C, D, E, ... (Next: F)

  • Skipping Letters:
    Example: A, C, E, G, ... (Skip one letter each time; Next: I)

  • Reverse Order:
    Example: Z, Y, X, W, ... (Next: V)

  • Patterned Increments:
    Example: A, D, G, J, ... (Add 3 positions each time; Next: M)

  • Alternating Series:
    Example: A, C, B, D, C, E, ... (Alternate between +2 and -1)

Tips for Solving Letter Series

  • Write the position numbers (A=1, B=2, ..., Z=26) for clarity.

  • Check for forward or backward jumps.

  • Look for alternating or repeating patterns.

3. Figure Series (Non-Verbal Series)

Figure series involve a sequence of images or shapes that follow a visual or spatial pattern. These are common in non-verbal reasoning tests.

Common Patterns in Figure Series

  • Rotation: Each figure is rotated by a fixed angle.

  • Reflection: Each figure is mirrored.

  • Addition/Removal: Elements are added or removed in each step.

  • Shading/Color Change: The shading or color alternates or follows a pattern.

  • Movement: Parts of the figure move in a certain direction.

Tips for Solving Figure Series

  • Observe changes in shape, size, position, or shading.

  • Note the direction and degree of rotation or reflection.

  • Look for repetition or alternation in the pattern.

Detailed Examples for Each Type

Number Series Examples

  1. Arithmetic Series:
    5, 10, 15, 20, 25, ... (Add 5 each time)

  2. Geometric Series:
    2, 6, 18, 54, ... (Multiply by 3 each time)

  3. Square Number Series:
    4, 9, 16, 25, ... (2², 3², 4², 5²)

  4. Cube Number Series:
    8, 27, 64, 125, ... (2³, 3³, 4³, 5³)

  5. Fibonacci Series:
    1, 1, 2, 3, 5, 8, 13, ... (Each term is the sum of previous two)

  6. Two-Stage Series:
    1, 3, 6, 10, 15, ... (Differences: 2, 3, 4, 5)

  7. Mixed Series:
    7, 16, 34, 70, 142, ... (Multiply by 2, add 2)

Letter Series Examples

  1. Consecutive:
    A, B, C, D, ... (Next: E)

  2. Skipping:
    B, D, F, H, ... (Next: J)

  3. Reverse:
    Z, X, V, T, ... (Next: R)

  4. Patterned Increment:
    A, D, G, J, ... (Next: M; +3 each time)

  5. Alternating:
    M, N, L, O, K, P, ... (Alternate +1, -2)

Figure Series Examples

  1. Rotation:
    A triangle rotates 90° clockwise each time.

  2. Addition:
    A square gains one dot in each step.

  3. Shading:
    A circle alternates between shaded and unshaded.

  4. Movement:
    A line moves from left to right in each figure.

  5. Combination:
    A shape rotates and changes color alternately.

  6. Number Series (1–50)

    1. 2, 4, 6, 8, ?

      • Answer: 10
        Explanation: Add 2 each time.

    2. 7, 10, 8, 11, 9, 12, ?

      • Answer: 10
        Explanation: Alternate +3, -2.

    3. 36, 34, 30, 28, 24, ?

      • Answer: 22
        Explanation: -2, -4, -2, -4, ...

    4. 3, 9, 27, 81, ?

      • Answer: 243
        Explanation: ×3 each time.

    5. 240, 60, 15, 5, ?

      • Answer: 1
        Explanation: ÷4, ÷4, ÷3, ÷5.

    6. 1, 4, 9, 16, ?

      • Answer: 25
        Explanation: Squares: 1², 2², 3², 4², 5².

    7. 8, 27, 64, 125, ?

      • Answer: 216
        Explanation: Cubes: 2³, 3³, 4³, 5³, 6³.

    8. 0, 1, 1, 2, 3, 5, ?

      • Answer: 8
        Explanation: Fibonacci series.

    9. 4, 9, 6, 11, 8, ?

      • Answer: 13
        Explanation: +5, -3 alternately.

    10. 12, 7, 21, 16, 30, ?

      • Answer: 25
        Explanation: -5, +14 alternately.

    11. 80, 10, 70, 15, 60, ?

      • Answer: 20
        Explanation: -70, +5 alternately.

    12. 8, 6, 9, 23, 87, ?

      • Answer: 445
        Explanation: ×1-2, ×2-3, ×3-4, ×4-5, ×5-6.

    13. 64, 58, 52, 46, ?

      • Answer: 40
        Explanation: -6 each time.

    14. 5, 12, 26, ?, 110, 222

      • Answer: 54
        Explanation: ×2+2, ×2+2, etc.

    15. 1, 2, 6, 15, ?, 56, 92

      • Answer: 31
        Explanation: ×1+1, ×2+2, ×3+3, etc.

    16. 3, 11, 27, 59, ?

      • Answer: 123
        Explanation: ×2+5 each time.

    17. 560, 280, 140, 70, ?

      • Answer: 35
        Explanation: ÷2 each time.

    18. 13, 14, 23, 48, 97, 178, ?

      • Answer: 299
        Explanation: +1, +9, +25, +49, +81, +121 (squares).

    19. 62, 74, 80, 86, 95, ?, 158

      • Answer: 113
        Explanation: +12, +6, +6, +9, +18, +45.

    20. 19, 38, 35, ?, 135, 810

      • Answer: 140
        Explanation: ×2-1, ×1+3, ×4, ×6.

    21. 1, 3, 6, 10, 15, ?

      • Answer: 21
        Explanation: +2, +3, +4, +5, +6.

    22. 5, 10, 20, 40, ?

      • Answer: 80
        Explanation: ×2 each time.

    23. 2, 6, 12, 20, ?

      • Answer: 30
        Explanation: Add consecutive even numbers: +4, +6, +8, +10.

    24. 21, 18, 15, 12, ?

      • Answer: 9
        Explanation: -3 each time.

    25. 120, 60, 30, 15, ?

      • Answer: 7.5
        Explanation: ÷2 each time.

    26. 3, 6, 12, 24, ?

      • Answer: 48
        Explanation: ×2 each time.

    27. 1, 8, 27, 64, ?

      • Answer: 125
        Explanation: Cubes: 1³, 2³, 3³, 4³, 5³.

    28. 13, 11, 9, 7, ?

      • Answer: 5
        Explanation: -2 each time.

    29. 5, 18, 70, 278, ?

      • Answer: 1110
        Explanation: ×3+3, ×4-2, ×5+8, ×6-10.

    30. 6, 11, 21, 36, 56, ?

      • Answer: 81
        Explanation: +5, +10, +15, +20, +25.

    31. 1, 4, 9, 16, 25, ?

      • Answer: 36
        Explanation: Squares: 1², 2², 3², 4², 5², 6².

    32. 2, 5, 10, 17, 26, ?

      • Answer: 37
        Explanation: +3, +5, +7, +9, +11.

    33. 100, 96, 88, 76, ?

      • Answer: 60
        Explanation: -4, -8, -12, -16.

    34. 1, 2, 4, 8, 16, ?

      • Answer: 32
        Explanation: ×2 each time.

    35. 1, 3, 7, 15, 31, ?

      • Answer: 63
        Explanation: ×2+1 each time.

    36. 0, 6, 24, 60, 120, ?

      • Answer: 210
        Explanation: ×1, ×2, ×3, ×4, ×5.

    37. 2, 3, 5, 8, 12, ?

      • Answer: 17
        Explanation: +1, +2, +3, +4, +5.

    38. 7, 14, 28, 56, ?

      • Answer: 112
        Explanation: ×2 each time.

    39. 10, 20, 40, 80, ?

      • Answer: 160
        Explanation: ×2 each time.

    40. 1, 4, 10, 22, 46, ?

      • Answer: 94
        Explanation: ×2+2, ×2+2, etc.

    41. 3, 6, 11, 18, ?

      • Answer: 27
        Explanation: +3, +5, +7, +9.

    42. 5, 10, 17, 26, ?

      • Answer: 37
        Explanation: +5, +7, +9, +11.

    43. 2, 4, 12, 48, ?

      • Answer: 240
        Explanation: ×2, ×3, ×4, ×5.

    44. 1, 2, 6, 24, ?

      • Answer: 120
        Explanation: ×1, ×2, ×3, ×4, ×5 (factorials).

    45. 3, 5, 11, 23, 43, 73, ?

      • Answer: 115
        Explanation: +2, +6, +12, +20, +30, +42.

    46. 4, 8, 16, 32, ?

      • Answer: 64
        Explanation: ×2 each time.

    47. 9, 18, 36, 72, ?

      • Answer: 144
        Explanation: ×2 each time.

    48. 6, 12, 24, 48, ?

      • Answer: 96
        Explanation: ×2 each time.

    49. 2, 5, 12, 29, ?

      • Answer: 70
        Explanation: ×2+1, ×2+2, ×2+3, ×2+4.

    50. 1, 3, 6, 10, 15, ?

      • Answer: 21
        Explanation: +2, +3, +4, +5, +6.

    Letter Series (51–80)

    1. A, C, E, G, ?

      • Answer: I
        Explanation: Skip one letter each time.

    2. M, N, O, P, ?

      • Answer: Q
        Explanation: Next consecutive letter.

    3. D, F, H, J, ?

      • Answer: L
        Explanation: Skip one letter.

    4. Z, X, V, T, ?

      • Answer: R
        Explanation: Skip one letter backward.

    5. B, E, H, K, ?

      • Answer: N
        Explanation: +3 each time.

    6. A, D, G, J, ?

      • Answer: M
        Explanation: +3 each time.

    7. C, F, I, L, ?

      • Answer: O
        Explanation: +3 each time.

    8. P, Q, S, V, ?

      • Answer: Z
        Explanation: +1, +2, +3, +4.

    9. A, Z, B, Y, ?

      • Answer: C
        Explanation: Alternate forward and backward.

    10. E, H, K, N, ?

      • Answer: Q
        Explanation: +3 each time.

    11. M, O, Q, S, ?

      • Answer: U
        Explanation: +2 each time.

    12. B, D, G, K, ?

      • Answer: P
        Explanation: +2, +3, +4, +5.

    13. F, I, M, R, ?

      • Answer: X
        Explanation: +3, +4, +5, +6.

    14. A, C, F, J, O, ?

      • Answer: U
        Explanation: +2, +3, +4, +5, +6.

    15. Q, R, T, W, ?

      • Answer: A
        Explanation: +1, +2, +3, +4 (wraps after Z).

    16. D, G, K, P, ?

      • Answer: V
        Explanation: +3, +4, +5, +6.

    17. M, L, J, G, ?

      • Answer: C
        Explanation: -1, -2, -3, -4.

    18. B, E, J, Q, ?

      • Answer: Z
        Explanation: +3, +5, +7, +9.

    19. C, F, J, O, ?

      • Answer: U
        Explanation: +3, +4, +5, +6.

    20. W, T, Q, N, ?

      • Answer: K
        Explanation: -3 each time.

    21. A, E, I, M, ?

      • Answer: Q
        Explanation: +4 each time.

    22. B, F, J, N, ?

      • Answer: R
        Explanation: +4 each time.

    23. C, H, N, U, ?

      • Answer: C
        Explanation: +5, +6, +7, (wraps after Z).

    24. M, P, S, V, ?

      • Answer: Y
        Explanation: +3 each time.

    25. Z, X, U, Q, ?

      • Answer: L
        Explanation: -2, -3, -4, -5.

    26. A, D, H, M, ?

      • Answer: S
        Explanation: +3, +4, +5, +6.

    27. B, C, E, H, L, ?

      • Answer: Q
        Explanation: +1, +2, +3, +4, +5.

    28. F, J, O, U, ?

      • Answer: B
        Explanation: +4, +5, +6, +7 (wraps after Z).

    29. D, H, M, S, ?

      • Answer: Z
        Explanation: +4, +5, +6, +7.

    30. G, K, P, V, ?

      • Answer: C
        Explanation: +4, +5, +6, +7 (wraps after Z).

    Figure Series (81–100)

    Note: Since figures cannot be visually displayed, these are described as clearly as possible.

    1. A triangle rotates 90° clockwise each time. What comes after 270°?

      • Answer: Original position (360°/0°).

    2. A square gains one dot in each step (1, 2, 3, 4, ?).

      • Answer: 5 dots.

    3. A circle alternates between shaded and unshaded. What comes after shaded?

      • Answer: Unshaded.

    4. A line moves from left to right in four steps. Where is it in step 5?

      • Answer: Back to left (cycle repeats).

    5. A figure is mirrored in each step. What comes after the mirrored image?

      • Answer: Original image.

    6. A square adds a triangle on top, then on the right, then on the bottom, then on the left. What comes next?

      • Answer: Triangle on top (cycle repeats).

    7. A pentagon changes color from red to blue to green to red. What comes next?

      • Answer: Blue.

    8. A star inside a circle moves clockwise to each corner. Where is it after four moves?

      • Answer: Back to original position.

    9. A figure increases in size each time. What comes after the largest?

      • Answer: Smallest (cycle restarts).

    10. A shape rotates 45° each time. What is the rotation after 360°?

      • Answer: 0° (original position).

    11. A rectangle alternates between horizontal and vertical orientation. What comes after vertical?

      • Answer: Horizontal.

    12. A dot moves from the top-left to top-right, then bottom-right, then bottom-left. What comes next?

      • Answer: Top-left.

    13. A figure is shaded in one quadrant, then the next, and so on. What comes after all four quadrants?

      • Answer: First quadrant (cycle repeats).

    14. A triangle doubles in size each time. What comes after the largest triangle?

      • Answer: Smallest triangle (cycle restarts).

    15. A circle inside a square moves outside the square, then back inside. What comes next?

      • Answer: Outside.

    16. A figure with one line, then two, then three, then four. What comes next?

      • Answer: Five lines.

    17. A pattern alternates between black and white stripes. What comes after white?

      • Answer: Black.

    18. A square splits into two, then four, then eight smaller squares. What comes next?

      • Answer: Sixteen small squares.

    19. A hexagon rotates 60° each time. What comes after 360°?

      • Answer: Original position.

    20. A circle is divided into halves, then quarters, then eighths. What comes next?
      - Answer: Sixteenth

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