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Polynomial Sequences

Polynomial Sequences follow quadratic (n²), cubic (n³), or higher-degree polynomial patterns. These sequences have constant second differences (for quadratic) or constant third differences (for cubic). These problems test your ability to identify higher-order patterns.

10Worksheets
200+Practice Questions
AdvancedDifficulty
3-4 hoursHours to Master

What You'll Learn

Introduction & Overview Why This Matters How to Solve Pro Tips & Tricks
Shortcut Methods Common Mistakes Practice Worksheets Exam Importance

Introduction to Polynomial Sequences

Polynomial Sequences follow quadratic (n²), cubic (n³), or higher-degree polynomial patterns. These sequences have constant second differences (for quadratic) or constant third differences (for cubic). These problems test your ability to identify higher-order patterns.

Prerequisites

Quadratic and cubic numbers Difference method Second differences Polynomial fitting
Why This Matters: Polynomial Sequences problems appear in 1-2 questions in Banking PO and SSC CGL exams. They test advanced pattern recognition.

How to Solve Polynomial Sequences Problems

1

Step 1: Calculate first differences between consecutive terms

2

Step 2: Calculate second differences (differences of first differences)

3

Step 3: If second differences are constant → quadratic sequence

4

Step 4: If third differences are constant → cubic sequence

5

Step 5: For next term: extend the difference pattern

6

Step 6: Verify the pattern holds for all given terms

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Step 7: Present the next term

Pro Strategy: Use the method of differences. Constant second differences indicate a quadratic pattern. Constant third differences indicate a cubic pattern. Extend the difference table to find the next term.

Example Problem

Example: Find the next term: 1, 4, 9, 16, 25, ___ Solution: Step 1: First differences: 3,5,7,9 Step 2: Second differences: 2,2,2 (constant) Step 3: Quadratic sequence (n²) Step 4: Next first difference = 9 + 2 = 11 Step 5: Next term = 25 + 11 = 36 Answer: 36

Pro Tips & Tricks

  • Quadratic sequence: aₙ = An² + Bn + C
  • Cubic sequence: aₙ = An³ + Bn² + Cn + D
  • Constant second differences → quadratic
  • Constant third differences → cubic
  • The nth term of a quadratic is n² + c or n(n+1)/2
  • Sum of first n natural numbers: n(n+1)/2 (quadratic)

Shortcut Methods to Solve Faster

For quadratic, next term = last term + last first difference + second difference
For cubic, next term = last term + last first diff + last second diff + third diff
Second difference = 2A (where A is coefficient of n²)
Common quadratic sequences: n², n²+1, n(n+1), 2n²-1

Common Mistakes to Avoid

Not calculating differences systematically
Assuming pattern is quadratic when differences not constant
Forgetting to extend all difference rows
Confusing quadratic with cubic patterns

Exam Importance

Polynomial Sequences is an important topic for various competitive exams. Here's how frequently it appears:

SSC CGL
1-2 questions
BANKING PO
1-2 questions
RAILWAYS RRB
1-2 questions
CAT
2-3 questions
INSURANCE
1-2 questions

Ready to Master Polynomial Sequences?

Start with Worksheet 1 and work your way up to expert level! Each worksheet includes:

20 practice questions
Detailed solutions
Step-by-step explanations
Start Practicing Now