The solutions for Chapter 12 – Mathematical Induction from RD Sharma Class 11 Maths are thoughtfully prepared to help students perform well in their board examinations. This chapter introduces students to the principles of mathematical induction, an essential concept for higher-level mathematics.
Our expert faculty has created step-by-step solutions that simplify complex topics, making it easier for students to grasp the logic and apply it efficiently during exams. These solutions are particularly beneficial for learners aiming to strengthen their conceptual understanding and improve their speed and accuracy in solving problems.
Chapter 12 includes two comprehensive exercises, each designed to reinforce the understanding of mathematical induction through practice-based learning. The chapter mainly covers the foundational principles and guides students through precise answers for each exercise, enhancing their problem-solving skills.
To explore the complete set of solutions and get exam-ready, students can access the latest version of RD Sharma Class 11 Chapter 12 solutions, updated for the 2025–26 academic session using the download links provided below.
RD Sharma Solutions Class 12 Maths Mathematical Induction : Highlights
- Understanding mathematical statements
- The principle of mathematical induction
- The first principle of mathematical induction
- The second principle of mathematical induction
These concepts build a strong base for advanced mathematical thinking and are crucial for competitive and board-level exam preparation.
RD Sharma Solutions Class 11 Maths Chapter 12 - Mathematical Induction Question with Answers
Q1. Prove: 1 + 2 + 3 + ⋯ + n = n(n+1)/2
Base Case (n = 1): LHS = 1, RHS = (1×2)/2 = 1 ✅
Inductive Step: Assume true for n = k: 1 + 2 + ⋯ + k = k(k+1)/2
Prove for k+1:
LHS = k(k+1)/2 + (k+1) = (k+1)(k+2)/2 ✅
Q2. Prove: 1² + 2² + ⋯ + n² = n(n+1)(2n+1)/6
Use base case, assume true for n = k, then prove for n = k+1 using algebra.
Q3. Prove: 1³ + 2³ + ⋯ + n³ = [n(n+1)/2]²
Use induction similarly to Q1 and Q2. Expand and match both sides.
Type 2: Inequalities via Induction
Q4. Prove: 2ⁿ > n for all n ≥ 1
Base: 2¹ = 2 > 1 ✅
Assume true for k: 2ᵏ > k
Then 2ᵏ⁺¹ = 2×2ᵏ > 2k ≥ k+1 for k ≥ 1 ✅
Q5. Prove: n! > 2ⁿ for all n ≥ 4
Base: 4! = 24 > 16 ✅
Inductive step: Assume true for k, prove for k+1 using factorial expansion.
Q6. Prove: 3ⁿ > n³ for all n ≥ 4
Prove using base case and increasing difference in growth rates.
Type 3: Divisibility Proofs
Q7. Prove: 5ⁿ − 1 is divisible by 4
Base: 5¹ − 1 = 4 ✅
Assume: 5ᵏ − 1 = 4m
Then: 5ᵏ⁺¹ − 1 = 5×5ᵏ − 1 = 5(4m + 1) − 1 = 20m + 4 = 4(5m + 1) ✅
Q8. Prove: 7ⁿ − 1 divisible by 6
Use binomial expansion or modular arithmetic under mod 6.
Q9. Prove: 11ⁿ − 4ⁿ divisible by 7
Inductive steps using assumed value and simplification.
Q10. Prove: 23n − 1 divisible by 7
Base case: 2³ − 1 = 7 ✅
Then use pattern or induction to prove for n = k+1.
Type 4: Algebraic and Inequality Statements
Q11. Prove: (1 + x)n ≥ 1 + nx for x > -1
Use binomial theorem and eliminate higher-order terms.
Q12. Prove: (1 + 1/n)n < 3 for n ≥ 1
Known convergence to e < 3. Use induction.
Type 5: Expression Forms and Factor Proofs
Q14. Prove: n³ + 2n divisible by 3
Check base case, then show divisibility holds under k+1.
Q15. Show: 4ⁿ + 15n − 1 divisible by 9
Base: n = 1 ⇒ 4 + 15 − 1 = 18 ✅
Q16. Show: 9ⁿ − 8n − 1 divisible by 64
Use substitution and modular arithmetic or expansion.
Advanced Problems (Q17–Q40)
Q17. Prove: n(n+1)(n+2) divisible by 6
Three consecutive numbers ⇒ always divisible by 2 and 3
Q18. Prove: 2ⁿ + 1 divisible by 3 when n is odd
Use mod 3 pattern
Q19. Fibonacci-style recurrence: F(n) = 2F(n−1) + 3F(n−2)
Use base cases and apply induction
Q20. Prove: Sum of first n odd numbers = n²
1 + 3 + 5 + … + (2n−1) = n² ✅
Q21. ∑k⁴ = n(n+1)(2n+1)(3n² + 3n −1)/30
Q22. ∑k(k+1) = n(n+1)(n+2)/3
Q23. (2n)! > 2ⁿ·n! for n ≥ 2
Q24. ∑1/[k(k+1)] = n/(n+1)
Q25. ∑k(k+1) = n(n+1)(n+2)/3
Q26. ∑(2k−1) = n²
Q27. ∑1/[(2k−1)(2k+1)] = n/(2n+1)
Q28. (n²+n)² = n⁴ + 2n³ + n²
Q29. n² < 2ⁿ for n ≥ 5
Q30. ∑k³ = [n(n+1)/2]²
Q31. n⁴ − n² divisible by 12 for n > 1
Q32. n⁵ − n divisible by 5
Q33. ∑k(k+1)(k+2) = n(n+1)(n+2)(n+3)/4
Q34. n³ − n divisible by 6
Q35. n³ + 2n divisible by 3
Q36. 4ⁿ + 15n − 1 divisible by 9
Q37. 2²ⁿ − 1 divisible by 3
Q38. 10ⁿ + 1 divisible by 11 for odd n
Q39. 7ⁿ + 2ⁿ divisible by 3
Q40. ∑1/[k(k+1)(k+2)] = n(n+3)/[4(n+1)(n+2)]
Frequently Asked Questions
While it’s a small chapter, it's not advisable to skip Mathematical Induction. Questions are generally easy and scoring. Also, understanding induction builds confidence in proof-based topics and is often a base for more advanced concepts in higher mathematics.
To master Mathematical Induction:
Understand the three-step process: base case, inductive hypothesis, and inductive step.
Practice all solved examples in RD Sharma Chapter 12.
Attempt a variety of problems (sums, inequalities, divisibility).
Watch explanatory videos and refer to step-by-step solutions.
Consistent practice and conceptual clarity are key to mastering this topic.
Questions in RD Sharma Chapter 12 are mostly structured around:
Summation Proofs (using known formulas)
Divisibility Proofs
Inequality Proofs
Mathematical Statements involving nth terms
Each question generally requires establishing a base case and then proving the inductive step.
You can find detailed, step-by-step RD Sharma Solutions for Chapter 12 on trusted educational platforms like Infinity Learn, or directly from your school’s recommended solution guides. These resources break down each question using the principle of mathematical induction with simplified explanations for easy understanding.
Yes, Mathematical Induction is important for foundational understanding and is often asked in JEE Main, CUET, and other entrance exams. While the weightage may be low, questions from this chapter can help boost scores in algebra-based problems. Mastery of the method also strengthens logical reasoning and mathematical proof skills.
RD Sharma Class 11 Maths Chapter 12 on Mathematical Induction typically includes 1 exercise (Exercise 12.1) with around 30–40 questions. These questions range from proving summation formulas to proving divisibility and inequalities using the principle of mathematical induction.