1. Class 11
2. Important Question for exams Class 11
3. Chapter 4 Class 11 Mathematical Induction

Transcript

Ex 4.1,13 Prove the following by using the principle of mathematical induction for all n โ N: ("1 + " 3/1) ("1 + " 5/4) ("1 + " 7/9)โฆ.. ("1 + " ((2๐ + 1))/๐2) = (n + 1)2 Let P(n) : ("1 + " 3/1) ("1 + " 5/4) ("1 + " 7/9)โฆ.. ("1 + " ((2๐ + 1))/๐2) = (n + 1)2 For n = 1, L.H.S = ("1 + " 3/1) = 1 + 3 = 4 R.H.S = (1 + 1)2 = 22 = 4 Thus, L.H.S. = R.H.S , โดP(n) is true for n = 1 Assuming P(k) is true P(k) : ("1 + " 3/1) ("1 + " 5/4) ("1 + " 7/9)โฆ.. ("1 + " ((2๐ + 1))/๐2) = (k + 1)2 We will prove P(k + 1) is true R.H.S = ((k + 1) + 1)2 L.H.S = ("1 + " 3/1) ("1 + " 5/4) ("1 + " 7/9)โฆ.. ("1 + " ((2(๐ + 1) + 1))/(๐ + 1)2) L.H.S = R.H.S โด P(k + 1) is true whenever P(k) is true. โด By the principle of mathematical induction, P(n) is true for n, where n is a natural number

Chapter 4 Class 11 Mathematical Induction

Class 11
Important Question for exams Class 11