1. Introduction to Number Series

A number series is a sequence of numbers arranged in a particular pattern or rule. Identifying the underlying pattern is essential to predict the next term(s) in the series.

Number series problems are commonly asked in aptitude tests, competitive exams, and logical reasoning assessments.

2. Types of Number Series

(A) Arithmetic Series

  • Each term increases or decreases by a constant difference (d).
  • General Form: a, a + d, a + 2d, a + 3d, ...
  • Example: 2, 5, 8, 11, 14, ... (d = 3)

(B) Geometric Series

  • Each term is obtained by multiplying or dividing the previous term by a constant ratio (r).
  • General Form: a, ar, ar², ar³, ...
  • Example: 3, 6, 12, 24, 48, ... (r = 2)

(C) Square & Cube Series

  • Terms are based on squares or cubes of numbers.
  • Square Series Example: 1, 4, 9, 16, 25, ... (1², 2², 3², ...)
  • Cube Series Example: 1, 8, 27, 64, 125, ... (1³, 2³, 3³, ...)

(D) Fibonacci Series

  • Each term is the sum of the two preceding terms.
  • General Form: F(n) = F(n-1) + F(n-2)
  • Example: 0, 1, 1, 2, 3, 5, 8, ...

(E) Prime Number Series

  • Consists of prime numbers (divisible only by 1 and themselves).
  • Example: 2, 3, 5, 7, 11, 13, ...

(F) Alternating Series

  • Two or more different patterns alternate in the series.
  • Example: 2, 4, 3, 6, 4, 8, 5, ... (Alternate +2 and -1)

(G) Mixed Series

  • Combines multiple operations (e.g., addition, multiplication, exponents).
  • Example: 5, 11, 24, 51, 106, ... (×2 + 1, ×2 + 2, ...)

3. Strategies to Solve Number Series

  1. Check for Common Differences - Compute differences between consecutive terms.
  2. Check for Common Ratios - Divide consecutive terms to see if the ratio is constant.
  3. Look for Squares/Cubes - Check if numbers are perfect squares or cubes.
  4. Check for Fibonacci Pattern - See if each term is the sum of previous terms.
  5. Identify Alternating Patterns - If two different patterns alternate, separate them.
  6. Use Trial & Error for Complex Series - Sometimes, multiple operations are applied.

4. Solved Examples

Example 1:

Series: 3, 7, 11, 15, 19, ?

Solution:
Difference between terms: 7-3=4, 11-7=4, 15-11=4, etc.
Pattern: Arithmetic series with d = 4.
Next term: 19 + 4 = 23.

Example 2:

Series: 2, 6, 18, 54, ?

Solution:
Ratio between terms: 6/2=3, 18/6=3, 54/18=3.
Pattern: Geometric series with r = 3.
Next term: 54 × 3 = 162.

Example 3:

Series: 1, 4, 9, 16, 25, ?

Solution:
Terms are 1², 2², 3², 4², 5².
Next term: 6² = 36.

Example 4:

Series: 0, 1, 1, 2, 3, 5, ?

Solution:
Fibonacci series: 0+1=1, 1+1=2, 1+2=3, 2+3=5.
Next term: 3 + 5 = 8.

5. Practice Problems

1. Find the next term: 5, 10, 20, 40, ?

2. Find the missing term: 12, 24, 48, ?, 192

3. Complete the series: 1, 8, 27, 64, ?

4. Predict the next number: 2, 5, 10, 17, 26, ?

5. Find the missing term: 3, 6, 9, 15, 24, ?

6. What comes next: 1, 3, 7, 15, 31, ?

7. Complete the series: 100, 81, 64, 49, 36, ?

8. Find the next term: 0, 4, 18, 48, 100, ?

9. Predict the missing number: 2, 3, 5, 7, 11, 13, ?

10. What comes next: 1, 2, 6, 24, 120, ?

11. Find the next term: 1, 4, 27, 256, ?

12. Complete the series: 3, 5, 9, 17, 33, ?

13. Predict the next number: 1, 1, 2, 3, 5, 8, 13, 21, ?

14. Find the missing term: 2, 6, 12, 20, 30, ?

15. What comes next: 1, 11, 21, 1211, 111221, ?

6. Conclusion

  • Number series problems require pattern recognition and logical reasoning.
  • Practice different types to improve speed and accuracy.
  • Common patterns include arithmetic, geometric, square/cube, Fibonacci, and mixed series.