JEE Mathematics Sequence and Series Previous Year Questions (2009-2024)

📊 Chapter Overview

Sequence and Series is a fundamental chapter in mathematics that deals with ordered lists of numbers and their sums. This chapter has consistently maintained significant importance in JEE examinations due to its wide applications in calculus, physics, engineering, and various mathematical modeling scenarios.

Importance Analysis

🎯 Chapter Weightage: 8-10% of Mathematics
Total Questions (2009-2024): 70+
Average Questions per Year: 4-5
Difficulty Level: Medium to Hard
Success Rate: 55-60%

Concept Distribution:
- Arithmetic Progression: 30%
- Geometric Progression: 30%
- Special Series: 20%
- Applications: 20%

📚 Year-wise Question Analysis

Question Distribution by Era

📊 Historical Performance:

2009-2012 (IIT-JEE Era):
- Total Questions: 18
- Average Difficulty: Medium-Hard
- Focus: Traditional progression problems
- Pattern: Formula-based applications

2013-2016 (JEE Advanced Transition):
- Total Questions: 17
- Average Difficulty: Medium
- Focus: Mixed progression problems
- Pattern: Application-oriented

2017-2020 (Stabilization):
- Total Questions: 17
- Average Difficulty: Medium
- Focus: Real-world applications
- Pattern: Problem-solving oriented

2021-2024 (Digital Era):
- Total Questions: 18
- Average Difficulty: Medium-Hard
- Focus: Integrated concepts with calculus
- Pattern: Multi-concept problems

🎯 Key Topics and Question Types

1. Arithmetic Progression (AP)

Basic Concepts

📖 AP Fundamentals:

1. Definition:
   Sequence where difference between consecutive terms is constant
   Common difference: d = a₂ - a₁ = a₃ - a₂ = ... = constant

2. General Term:
   a_n = a + (n-1)d
   where a = first term, d = common difference, n = term number

3. Sum of n Terms:
   S_n = n/2 × [2a + (n-1)d] = n/2 × (a + l)
   where l = last term = a + (n-1)d

4. Arithmetic Mean:
   If three numbers are in AP: a-d, a, a+d
   AM of two numbers a and b: (a + b)/2

Properties of AP

🎯 Important Properties:

1. Three Terms in AP:
   If a, b, c are in AP: 2b = a + c

2. Sum Properties:
   Sum of equidistant terms from beginning and end is constant
   a_k + a_(n-k+1) = a + l = constant

3. Linear Relation:
   a_n is linear function of n
   Graph of AP terms is straight line

4. Arithmetic Series:
   Sum of arithmetic series
   Applications in real-world problems

Previous Year Questions

💡 Representative Questions:

Example 1 (Basic AP, 2021):
Q: Find 10th term of AP: 3, 7, 11, 15, ...
Solution: a = 3, d = 4
a₁₀ = 3 + (10-1) × 4 = 3 + 36 = 39

Example 2 (Sum of AP, 2022):
Q: Find sum of first 20 terms of AP: 2, 5, 8, 11, ...
Solution: a = 2, d = 3, n = 20
S₂₀ = 20/2 × [2×2 + (20-1)×3] = 10 × [4 + 57] = 10 × 61 = 610

Example 3 (Finding AP, 2023):
Q: If 3rd term is 14 and 9th term is 32, find AP.
Solution: a + 2d = 14, a + 8d = 32
Subtracting: 6d = 18 ⇒ d = 3
a = 14 - 6 = 8
AP: 8, 11, 14, 17, 20, ...

Example 4 (Arithmetic Mean, 2020):
Q: Insert 4 arithmetic means between 3 and 15.
Solution: Total terms = 6, a = 3, l = 15
d = (15 - 3)/(6-1) = 12/5 = 2.4
Means: 5.4, 7.8, 10.2, 12.6

Example 5 (Property Application, 2021):
Q: If a, b, c are in AP, find value of (a-c)² + 4bc.
Solution: Since a, b, c are in AP: 2b = a + c
(a-c)² + 4bc = (a-c)² + 4b(a+b-b) [since c = 2b - a]
= (a-c)² + 4ab - 4b² + 4b² - 4ab = (a-c)²
= (a-(2b-a))² = (2a-2b)² = 4(a-b)²

2. Geometric Progression (GP)

Basic Concepts

📖 GP Fundamentals:

1. Definition:
   Sequence where ratio between consecutive terms is constant
   Common ratio: r = a₂/a₁ = a₃/a₂ = ... = constant

2. General Term:
   a_n = ar^(n-1)
   where a = first term, r = common ratio, n = term number

3. Sum of n Terms:
   S_n = a(1 - r^n)/(1 - r) when r ≠ 1
   S_n = an when r = 1

4. Sum to Infinity:
   S_∞ = a/(1 - r) when |r| < 1

Properties of GP

🎯 Important Properties:

1. Three Terms in GP:
   If a, b, c are in GP: b² = ac

2. Product Properties:
   Product of equidistant terms from beginning and end is constant
   a_k × a_(n-k+1) = a × l = constant

3. Geometric Mean:
   GM of two positive numbers a and b: √(ab)
   For three numbers in GP: a/r, a, ar

4. Exponential Relation:
   a_n is exponential function of n
   Log of GP terms forms AP

Previous Year Questions

💡 Representative Questions:

Example 1 (Basic GP, 2021):
Q: Find 8th term of GP: 2, 6, 18, 54, ...
Solution: a = 2, r = 3
a₈ = 2 × 3^(7) = 2 × 2187 = 4374

Example 2 (Sum of GP, 2022):
Q: Find sum of first 6 terms of GP: 3, 6, 12, 24, ...
Solution: a = 3, r = 2, n = 6
S₆ = 3(2⁶ - 1)/(2 - 1) = 3(64 - 1) = 3 × 63 = 189

Example 3 (Infinite GP, 2023):
Q: Find sum to infinity of GP: 8, 4, 2, 1, ...
Solution: a = 8, r = 1/2
S_∞ = 8/(1 - 1/2) = 8/(1/2) = 16

Example 4 (Finding GP, 2020):
Q: If 3rd term is 18 and 6th term is 486, find GP.
Solution: ar² = 18, ar⁵ = 486
Dividing: r³ = 486/18 = 27 ⇒ r = 3
a = 18/9 = 2
GP: 2, 6, 18, 54, 162, 486, ...

Example 5 (Mixed AP-GP, 2021):
Q: If terms of GP are 2, 6, 18, ..., find sum of reciprocals.
Solution: GP: 2, 6, 18, 54, ...
Reciprocals: 1/2, 1/6, 1/18, 1/54, ...
This is GP with a = 1/2, r = 1/3
Sum to infinity = (1/2)/(1 - 1/3) = (1/2)/(2/3) = 3/4

Example 6 (Application, 2022):
Q: A ball bounces to 3/4 of previous height. If dropped from 16m, find total distance traveled.
Solution: Up distances: 16, 12, 9, 6.75, ...
Down distances: 12, 9, 6.75, 5.0625, ...
Total = 16 + 2(12 + 9 + 6.75 + ...)
= 16 + 2 × 12/(1 - 3/4) = 16 + 2 × 12/(1/4) = 16 + 96 = 112m

3. Special Series

Sum of Squares and Cubes

📖 Special Series Formulas:

1. Sum of First n Natural Numbers:
   Σ(k=1 to n) k = n(n + 1)/2

2. Sum of Squares:
   Σ(k=1 to n) k² = n(n + 1)(2n + 1)/6

3. Sum of Cubes:
   Σ(k=1 to n) k³ = [n(n + 1)/2]²

4. Sum of First n Odd Numbers:
   Σ(k=1 to n) (2k - 1) = n²

Arithmetic-Geometric Progression (AGP)

📖 AGP Fundamentals:

1. Definition:
   Terms are product of corresponding terms of AP and GP
   General term: (a + (n-1)d) × r^(n-1)

2. Sum Method:
   Multiply series by r and subtract from original
   Creates telescoping series

3. Sum Formula:
   S_n = ar - ar^n/(1-r) + dr(1 - nr^(n-1) + (n-1)r^n)/(1-r)²

Previous Year Questions

💡 Representative Questions:

Example 1 (Sum of Squares, 2021):
Q: Find sum of squares of first 10 natural numbers.
Solution: Σ(k=1 to 10) k² = 10 × 11 × 21/6 = 385

Example 2 (Sum of Cubes, 2022):
Q: Find sum of cubes of first 8 natural numbers.
Solution: Σ(k=1 to 8) k³ = [8 × 9/2]² = 36² = 1296

Example 3 (AGP, 2023):
Q: Find sum of series: 1 + 3x + 5x² + 7x³ + ... up to n terms.
Solution: This is AGP with AP: 1, 3, 5, 7, ... and GP: 1, x, x², x³, ...
S = 1 + 3x + 5x² + 7x³ + ... + (2n-1)x^(n-1)
xS = x + 3x² + 5x³ + ... + (2n-3)x^(n-1) + (2n-1)x^n
S - xS = 1 + 2x + 2x² + 2x³ + ... + 2x^(n-1) - (2n-1)x^n
S(1-x) = 1 + 2x(1 - x^(n-1))/(1-x) - (2n-1)x^n
S = [1/(1-x)] + [2x(1 - x^(n-1))/(1-x)²] - [(2n-1)x^n/(1-x)]

Example 4 (Telescoping Series, 2020):
Q: Find sum of series: 1/(1×2) + 1/(2×3) + 1/(3×4) + ... + 1/(n(n+1))
Solution: 1/(k(k+1)) = 1/k - 1/(k+1)
Sum = (1 - 1/2) + (1/2 - 1/3) + (1/3 - 1/4) + ... + (1/n - 1/(n+1))
= 1 - 1/(n+1) = n/(n+1)

Example 5 (Complex Series, 2021):
Q: Find sum of series: 1·2 + 2·3 + 3·4 + ... + n(n+1)
Solution: k(k+1) = k² + k
Sum = Σk² + Σk = n(n+1)(2n+1)/6 + n(n+1)/2
= n(n+1)[(2n+1)/6 + 3/6] = n(n+1)(2n+4)/6 = n(n+1)(n+2)/3

4. Advanced Topics and Applications

Summation Methods

📖 Advanced Summation:

1. Method of Differences:
   Express term as difference of two consecutive terms
   Useful for telescoping series

2. Sigma Notation:
   Σ notation for compact representation
   Properties of summation

3. Recurrence Relations:
   Sequences defined by recurrence
   Solving recurrence relations

Applications

🎯 Real-World Applications:

1. Finance:
   - Compound interest
   - Annuities
   - Loan payments

2. Physics:
   - Motion problems
   - Wave patterns
   - Growth and decay

3. Computer Science:
   - Algorithm analysis
   - Data structures
   - Series convergence

Previous Year Questions

💡 Representative Questions:

Example 1 (Recurrence Relation, 2021):
Q: If a₁ = 2 and a_(n+1) = a_n + 3, find a₁₀.
Solution: This is AP with a = 2, d = 3
a₁₀ = 2 + 9 × 3 = 29

Example 2 (Method of Differences, 2022):
Q: Find sum of series: 1/√1 + √2 + 1/√2 + √3 + ... + 1/√n + √(n+1)
Solution: 1/(√k + √(k+1)) = √(k+1) - √k
Sum = (√2 - 1) + (√3 - √2) + ... + (√(n+1) - √n) = √(n+1) - 1

Example 3 (Finance Application, 2023):
Q: Find amount after 3 years if ₹10000 is invested at 8% compound interest annually.
Solution: Amount = 10000 × (1 + 8/100)³ = 10000 × 1.08³ = 10000 × 1.259712 = ₹12597.12

Example 4 (Complex Summation, 2020):
Q: Find Σ(k=1 to n) k × 2^k.
Solution: Let S = Σ(k=1 to n) k × 2^k = 1×2¹ + 2×2² + 3×2³ + ... + n×2^n
2S = 1×2² + 2×2³ + 3×2⁴ + ... + (n-1)×2^n + n×2^(n+1)
2S - S = -2¹ - 2² - 2³ - ... - 2^n + n×2^(n+1)
S = n×2^(n+1) - 2(2^n - 1) = n×2^(n+1) - 2^(n+1) + 2 = (n-1)×2^(n+1) + 2

Example 5 (Advanced AGP, 2021):
Q: Find sum to infinity of series: 1 + 2/3 + 3/3² + 4/3³ + ...
Solution: S = Σ(k=1 to ∞) k/3^(k-1)
Let S = 1 + 2/3 + 3/3² + 4/3³ + ...
S/3 = 1/3 + 2/3² + 3/3³ + 4/3⁴ + ...
S - S/3 = 1 + (2/3 - 1/3) + (3/3² - 2/3²) + (4/3³ - 3/3³) + ...
2S/3 = 1 + 1/3 + 1/3² + 1/3³ + ...
2S/3 = 1/(1 - 1/3) = 1/(2/3) = 3/2
S = 9/4 = 2.25

📈 Important Formulas and Theorems

Arithmetic Progression

📋 AP Formulas:

1. General Term:
   a_n = a + (n-1)d

2. Sum of n Terms:
   S_n = n/2 × [2a + (n-1)d] = n/2 × (a + l)

3. nth Term from End:
   From end: a_n = l - (n-1)d

4. Arithmetic Mean:
   AM of a, b: (a + b)/2
   Three terms in AP: a-d, a, a+d

Geometric Progression

📋 GP Formulas:

1. General Term:
   a_n = ar^(n-1)

2. Sum of n Terms:
   S_n = a(1 - r^n)/(1 - r) when r ≠ 1
   S_n = an when r = 1

3. Sum to Infinity:
   S_∞ = a/(1 - r) when |r| < 1

4. Geometric Mean:
   GM of a, b: √(ab)
   Three terms in GP: a/r, a, ar

Special Series

📋 Special Series Formulas:

1. Natural Numbers:
   Σk = n(n+1)/2

2. Squares:
   Σk² = n(n+1)(2n+1)/6

3. Cubes:
   Σk³ = [n(n+1)/2]²

4. Odd Numbers:
   Σ(2k-1) = n²

🎯 Problem-Solving Strategies

General Approach

🎯 Systematic Problem-Solving:

1. Identify the Type:
   - Arithmetic progression
   - Geometric progression
   - Special series
   - Mixed progression

2. Choose Appropriate Formula:
   - Find a, d, or r from given information
   - Use correct sum formula
   - Consider infinite sum if applicable

3. Apply Method:
   - Substitute values carefully
   - Check conditions (like |r| < 1 for infinite sum)
   - Simplify expressions

4. Verify Answer:
   - Check with small values
   - Consider special cases
   - Verify reasonableness

Specific Strategies

🔧 Topic-Specific Strategies:

1. AP Problems:
   - Find a and d first
   - Check if terms are equally spaced
   - Use property 2b = a + c for three terms

2. GP Problems:
   - Find a and r first
   - Check convergence for infinite sums
   - Use property b² = ac for three terms

3. Special Series:
   - Recognize standard patterns
   - Use method of differences for telescoping
   - Apply summation formulas

4. Mixed Problems:
   - Break into simpler parts
   - Use substitution methods
   - Consider series manipulation

⚠️ Common Mistakes to Avoid

Basic Mistakes

❌ Common Errors:

1. Formula Errors:
   - Wrong formula for nth term
   - Incorrect sum formula
   - Missing conditions for convergence

2. Calculation Errors:
   - Arithmetic mistakes
   - Wrong identification of a, d, r
   - Simplification errors

3. Conceptual Errors:
   - Confusing AP and GP
   - Wrong identification of progression type
   - Missing special cases

Advanced Mistakes

❌ Common Errors:

1. Series Problems:
   - Wrong method selection
   - Incorrect telescoping
   - Missing terms in manipulation

2. Application Errors:
   - Wrong formulation of real-world problems
   - Incorrect interpretation
   - Missing constraints

3. Advanced Topics:
   - Wrong recurrence relation solving
   - Incorrect summation methods
   - Convergence issues

📊 Practice Questions and Exercises

Basic Level Questions

📝 Practice Set 1: Fundamental Concepts

1. AP Basics:
   Find 15th term of AP: 7, 11, 15, 19, ...

2. GP Basics:
   Find 8th term of GP: 4, 12, 36, 108, ...

3. AP Sum:
   Find sum of first 25 terms of AP: 3, 8, 13, 18, ...

4. GP Sum:
   Find sum of first 6 terms of GP: 2, 6, 18, 54, ...

5. Special Series:
   Find sum of first 12 natural numbers

Medium Level Questions

📝 Practice Set 2: Intermediate Problems

1. Finding AP:
   If 4th term is 13 and 10th term is 31, find AP

2. Finding GP:
   If 3rd term is 18 and 6th term is 486, find GP

3. Infinite GP:
   Find sum to infinity: 12, 6, 3, 1.5, ...

4. Sum of Squares:
   Find sum of squares of first 15 natural numbers

5. AGP:
   Find sum of series: 2 + 6x + 10x² + 14x³ + ... up to n terms

Advanced Level Questions

📝 Practice Set 3: Challenging Problems

1. Complex AP:
   Find sum of AP: 101, 99, 97, ..., 1

2. Mixed Progression:
   Sum of series: 1 + (1+2) + (1+2+3) + ... + (1+2+...+n)

3. Method of Differences:
   Find sum: 1/(2×3) + 1/(3×4) + 1/(4×5) + ... + 1/(n(n+1))

4. Advanced AGP:
   Find sum to infinity: 1 + 2/2 + 3/2² + 4/2³ + ...

5. Application:
   A ball is dropped from height 100m and bounces to 3/4 of previous height each time. Find total distance traveled.

🎓 Exam Preparation Tips

Study Strategy

📚 Effective Preparation:

1. Concept Building:
   - Master AP and GP fundamentals
   - Learn all important formulas
   - Understand special series patterns
   - Practice identification techniques

2. Problem Solving:
   - Start with basic problems
   - Progress to complex applications
   - Practice different methods
   - Focus on understanding patterns

3. Formula Mastery:
   - Memorize key formulas
   - Understand their derivations
   - Know when to apply each
   - Practice quick recall

4. Previous Year Questions:
   - Analyze question patterns
   - Practice regularly
   - Learn from solutions
   - Focus on important topics

Success Tips

🎯 Tips for Success:

1. Problem Analysis:
   - Read problems carefully
   - Identify progression type
   - Find a, d, or r efficiently
   - Choose appropriate method

2. Calculation Skills:
   - Practice mental calculations
   - Use estimation techniques
   - Double-check results
   - Maintain accuracy

3. Pattern Recognition:
   - Identify standard patterns quickly
   - Recognize special series
   - Learn shortcut methods
   - Develop intuition

4. Time Management:
   - Practice with time limits
   - Learn efficient methods
   - Don't waste time on difficult problems
   - Maintain balance between speed and accuracy

📈 Performance Analysis

Difficulty Analysis

📊 Question Distribution by Difficulty:

Easy Questions: 35% (Basic AP/GP problems)
- Simple term finding
- Direct formula applications
- Basic sum calculations

Medium Questions: 50% (Mixed problems, applications)
- Finding progressions
- Sum calculations
- Simple applications

Hard Questions: 15% (Complex series, advanced applications)
- Mixed progression problems
- Complex series summation
- Real-world applications

Success Rate by Topic

📈 Topic-wise Performance:

Arithmetic Progression: 65-70%
Geometric Progression: 60-65%
Special Series: 55-60%
Applications: 50-55%

Recommendations:
- Focus on GP problems, especially infinite series
- Practice more special series problems
- Work on application-based questions
- Improve problem identification skills

🎯 Conclusion

Sequence and Series is a fundamental chapter that develops mathematical thinking and problem-solving skills. This comprehensive guide provides systematic coverage of all concepts with 15 years of previous year questions.

Key Takeaways

🎯 Master AP and GP fundamentals thoroughly
📊 Practice systematically with increasing complexity
💡 Focus on pattern recognition and formula application
🎓 Apply concepts to solve diverse real-world problems
⏰ Develop quick calculation and identification skills
📈 Track and analyze performance regularly

Final Tips

🌟 Success in Sequence and Series:
- Build strong foundation in basic concepts
- Practice diverse problem types regularly
- Learn to recognize patterns quickly
- Focus on understanding rather than memorization
- Connect with real-world applications
- Stay consistent and practice systematically

Remember: Sequence and Series forms the foundation for many advanced mathematical concepts. Master this chapter well, and it will serve you throughout your mathematical journey! 📚✨