# Ex 4.1, 11

Last updated at May 29, 2018 by Teachoo

Last updated at May 29, 2018 by Teachoo

Transcript

Ex 4.1,11 Prove the following by using the principle of mathematical induction for all n ∈ N: 11.2.3 + 12.3.4 + 13.4.5 + ……..+ 1𝑛 𝑛 + 1(𝑛 + 2) = 𝑛(𝑛 + 3)4(𝑛 + 1)(𝑛 + 2) Let P (n) : 11.2.3 + 12.3.4 + 13.4.5 + ……..+ 1𝑛 𝑛 + 1(𝑛 + 2) = 𝑛(𝑛 + 3)4(𝑛 + 1)(𝑛 + 2) For n = 1, L.H.S = 11.2.3 = 16 R.H.S = 1.(1 + 3)4(1 + 1)(1 + 2) = 1.44.2.3 = 12.3 = 16 Hence, L.H.S. = R.H.S , ∴ P(n) is true for n = 1 Assume P(k) is true 11.2.3 + 12.3.4 + 13.4.5 + ……..+ 1𝑘 𝑘 + 1(𝑘 + 2) = 𝑘(𝑘 + 3)4(𝑘 + 1)(𝑘 + 2) We will prove that P(k + 1) is true. R.H.S = 𝑘 + 1 𝑘 + 1+ 34 𝑘 + 1+ 1 𝑘 + 1+ 2 L.H.S = 11.2.3 + 12.3.4 + 13.4.5 + ……..+ 1 𝑘 + 1 𝑘 + 1+ 1( 𝑘 + 1+ 2) Rough Factorizing 𝐤𝟑+ 𝟒+ 𝟗𝐤+ 𝟔𝐤𝟐 Let f(k) = k3+ 4+ 9k+ 6k2 Putting k = − 1 f(−1) = (−1)3+ 4+9 −1+6 (−1)2 = −1 + 4− 9+6 = 0 Thus, (k + 1) is a factor of f(k) ∴ f(k) = (𝑘+1)( 𝑘2+5𝑘+4)

Class 11

Important Question for exams Class 11

- Chapter 1 Class 11 Sets
- Chapter 2 Class 11 Relations and Functions
- Chapter 3 Class 11 Trigonometric Functions
- Chapter 4 Class 11 Mathematical Induction
- Chapter 5 Class 11 Complex Numbers
- Chapter 6 Class 11 Linear Inequalities
- Chapter 7 Class 11 Permutations and Combinations
- Chapter 8 Class 11 Binomial Theorem
- Chapter 9 Class 11 Sequences and Series
- Chapter 10 Class 11 Straight Lines
- Chapter 11 Class 11 Conic Sections
- Chapter 12 Class 11 Introduction to Three Dimensional Geometry
- Chapter 13 Class 11 Limits and Derivatives
- Chapter 14 Class 11 Mathematical Reasoning
- Chapter 15 Class 11 Statistics
- Chapter 16 Class 11 Probability

About the Author

Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 8 years. He provides courses for Maths and Science at Teachoo. You can check his NCERT Solutions from Class 6 to 12.