Sequences and Series

1. What It Is

  • Sequences and Series questions test your ability to identify patterns in numbers, alphabets, or symbols and continue the sequence or find a missing element.
  • A sequence is an arranged set of numbers, letters, or objects in a specific order.
  • A series is the sum of sequence terms or progression of numbers.

Examples:

  • Numeric sequence: 2, 4, 6, 8, ? → next = 10
  • Alphabet sequence: A, C, E, G, ? → next = I
  • Mixed sequence: 1, 2, 4, 8, ? → next = 16

2. How to Approach

  1. Observe the given terms carefully – numbers, letters, positions.
  2. Identify the pattern – addition, subtraction, multiplication, division, squares, cubes, alternation, skipping, etc.
  3. Test the pattern with first few terms.
  4. Apply the pattern to find the missing term or next term.
  5. Double-check – see if all given terms follow the same rule.

3. Rules & Key Points

  • Common Patterns:
    • Arithmetic → +, –
    • Geometric → ×, ÷
    • Square / Cube → n², n³
    • Alternating → different operations in turn
    • Mixed sequences → combination of above
  • Letter Sequences: Use alphabet positions (A=1, B=2, …)
  • Coding symbols → use positions in ASCII or given order
  • Check carefully for alternating patterns

4. Tricks & Shortcuts

  1. Check differences between terms → arithmetic sequence
  2. Check ratios → geometric sequence
  3. Look for squares/cubes → 1, 4, 9, 16…
  4. Check alternating patterns → +2, ×2, +3, ×2 …
  5. Write sequence positions if letters or symbols → helps find pattern

5. Types of Questions

  1. Find the next term – continue the sequence.
  2. Find missing term – sequence with one or more missing terms.
  3. Sum of series – arithmetic or geometric series.
  4. Coding sequences – letters or symbols.
  5. Mixed / Complex sequences – alternating patterns, multiple operations.

6. Stepwise Solving Strategy

  1. Observe first few terms → note differences or ratios.
  2. Check pattern type → arithmetic, geometric, square/cube, alternating.
  3. Apply pattern rule → find missing/next term.
  4. Verify entire sequence → ensure pattern holds.
  5. Solve related questions → sum, position, or coded term.

7. Easy Practice Questions

1. Numeric Sequence

Sequence: 2, 4, 6, 8, ?
Solution:

  • Difference between terms = +2 → next term = 10 ✅
  • Answer: 10

2. Alphabet Sequence

Sequence: A, C, E, G, ?
Solution:

  • Positions: 1, 3, 5, 7 → +2 each
  • Next = 9 → I ✅
  • Answer: I

3. Mixed Sequence

Sequence: 1, 2, 4, 8, ?
Solution:

  • Multiply by 2 → 1×2=2, 2×2=4, 4×2=8
  • Next = 8×2 = 16 ✅
  • Answer: 16

4. Alternating Sequence

Sequence: 2, 4, 8, 10, 20, ?
Solution:

  • Pattern: ×2, +2 alternately → 2×2=4, 4+4=8? wait check
  • Let’s see: 2→4 (×2), 4→8 (×2?), 8→10 (+2), 10→20 (×2) → Pattern: ×2, +2, ×2
  • Next: 20 + 2 = 22 ✅
  • Answer: 22

5. Series Sum

Series: 2 + 4 + 6 + 8 + … + 20
Solution:

  • Arithmetic series → first term a=2, last term l=20, n=10 terms, difference d=2
  • Sum = n/2 × (a + l) = 10/2 × (2 + 20) = 5 × 22 = 110 ✅
  • Answer: 110

8. Difficult Practice Questions

6. Missing Term

Sequence: 3, 6, 12, ?, 48
Solution:

  • Pattern: ×2 → 3×2=6, 6×2=12, 12×2=24, 24×2=48 ✅
  • Answer: 24

7. Alternating Operations

Sequence: 2, 4, 12, 14, 42, ?
Solution:

  • Pattern: ×2, +2 alternately → 2×2=4, 4×3? wait
  • Check: 2→4 (+2)? 4→12 (×3) 12→14 (+2) 14→42 (×3) → Pattern: ×2? not consistent, ×3 works
  • Next: 42 + 2 = 44 ✅
  • Answer: 44

8. Alphabet + Numeric

Sequence: A1, B2, C3, ?
Solution:

  • Letters: A→B→C → next = D
  • Numbers: 1→2→3 → next = 4
  • Answer: D4

9. Square / Cube Pattern

Sequence: 1, 4, 9, 16, ?
Solution:

  • Pattern: squares → 1², 2², 3², 4² → next 5² = 25 ✅
  • Answer: 25

10. Complex Alternating

Sequence: 5, 10, 8, 16, 14, ?
Solution:

  • Pattern: +5, –2, +8, –2, +? → check
  • Sequence: 5→10 (+5), 10→8 (–2), 8→16 (+8), 16→14 (–2), next → 14 +16? wait pattern seems +double previous addition?
  • Pattern alternates: +5, –2, +8, –2, +? → see addition doubles: 5→8→16 → 2×?
  • Next: +16 → 14 +16=30 ✅
  • Answer: 30

✅ Quick Tips

  1. Look for differences or ratios first.
  2. Check for alternating operations.
  3. Use squares, cubes, positions for numeric/alphabet sequences.
  4. For series sum, use arithmetic or geometric formula.

For complex sequences, write positions and check pattern stepwise.