Number Series

A number series is a sequence of numbers arranged in a particular pattern. You are expected to identify the logic or rule behind the pattern and find either the missing number or the next number.

🧠 Objective: Analyze the pattern → Apply logic → Predict the next/missing number

🔹 Types of Number Series

Let’s explore common types of number series, their logic, and shortcut tips:

1. 👉 Arithmetic Series
   Definition: Each term is obtained by adding or subtracting a constant number (common difference).
   Formula: an = a + (n – 1) × d
   Example: 3, 6, 9, 12, ? (Add 3 each time) → Next: 15

2. 👉 Geometric Series
   Definition: Each term is obtained by multiplying/dividing the previous term by a constant (common ratio).
   Formula: an = a × r(n–1)
   Example: 2, 4, 8, 16, ? (×2 each time) → Next: 32

3. 👉 Square/Cube Series
   Definition: Terms are squares or cubes of natural numbers.
   Example: 1, 4, 9, 16, ? → Next: 25 (since 1², 2², 3², 4²…)

4. 👉 Alternating Series
   Definition: Pattern alternates between positions.
   Example: 1, 4, 2, 5, 3, ? → Next: 6 (alternating between +3 and -2)

5. 👉 Mixed Operations Series
   Definition: Combination of operations like +2, ×2, –3, etc.
   Example: 2, 4, 7, 11, 16, ? → +2, +3, +4, +5 → Next: +6 = 22

6. 👉 Fibonacci Series
   Definition: Each term is the sum of the two previous terms.
   Example: 0, 1, 1, 2, 3, 5, 8, ? → Next: 13

7. 👉 Prime Number Series
   Definition: Series involving prime numbers.
   Example: 2, 3, 5, 7, 11, 13, ? → Next: 17

8. 👉 Pattern in Differences
   Definition: Differences between numbers follow a pattern.
   Example: 2, 5, 10, 17, 26, ?
   Differences: +3, +5, +7, +9 → Next difference: +11 → 26 + 11 = 37

🔹 Common Shortcut Tips

• Check the difference between consecutive terms first.

• If differences vary consistently, it may be an arithmetic or quadratic series.

• Try dividing if numbers are increasing rapidly.

• Check for alternating patterns.

• Look for squares, cubes, or prime numbers.

🔹 Important Formulas

1. Sum of first n natural numbers:
   S = n(n + 1)/2

2. Sum of squares:
   S = n(n + 1)(2n + 1)/6

3. Sum of cubes:
   S = [n(n + 1)/2]²

🔹 Examples with Solutions

🧩 Example 1:
Find the missing number: 7, 14, 28, 56, ?

Solution: Multiply by 2 each time → Next: 56 × 2 = 112

🧩 Example 2:
Find the missing number: 4, 9, 16, 25, ?

Solution: These are square numbers → 2², 3², 4², 5² → Next: 6² = 36

🧩 Example 3:
Find the next number: 2, 6, 12, 20, 30, ?

Differences: +4, +6, +8, +10 → Next: +12 → 30 + 12 = 42

🧩 Example 4:
Find the missing number: 81, 27, 9, 3, ?

Dividing by 3 each time → Next: 3 ÷ 3 = 1

🧩 Example 5:
What comes next: 1, 1, 2, 3, 5, 8, ?

Fibonacci Series → Next: 5 + 8 = 13

🔹 Practice MCQs

🧠 1. What is the next number in the series: 3, 6, 11, 18, 27, ?

A. 36
B. 38
C. 37
D. 39

✅ Answer: C. 37
Explanation: Differences: +3, +5, +7, +9, → Next: +11 → 27 + 10 = 37

🧠 2. What is the missing number: 125, 64, 27, 8, ?

A. 0
B. 3
C. 1
D. 1.5

✅ Answer: C. 1
Explanation: Cubes in reverse: 5³, 4³, 3³, 2³ → Next: 1³ = 1

🧠 3. What is the missing number: 1, 4, 9, 16, 25, ?

A. 30
B. 36
C. 32
D. 40

✅ Answer: B. 36
Explanation: Square series: 1², 2², 3², 4², 5² → Next: 6² = 36

🧠 4. Complete the series: 2, 3, 5, 7, 11, 13, ?

A. 15
B. 17
C. 18
D. 19

✅ Answer: B. 17
Explanation: Prime number series → Next prime after 13 is 17