1. Chapter 4 Class 11 Mathematical Induction
2. Serial order wise
3. Ex 4.1

Transcript

Ex 4.1,6: Prove the following by using the principle of mathematical induction for all n โ N: 1.2 + 2.3 + 3.4 +โฆโฆ + n. (n + 1) = (๐(๐ + 1)(๐ + 2))/3 Let P(n): 1.2 + 2.3 + 3.4 +โฆ.+ n.(n + 1) = (๐(๐ + 1)(๐ + 2))/3 For n = 1, L.H.S = 1.2 = 2 R.H.S = (1(1+1)(1+2))/3 = 1.2.3/3 = 2 L.H.S. = R.H.S โด P(n) is true for n = 1 Assume P(k) be true 1.2 + 2.3 + 3.4 +โฆ.+ k.(k + 1) = (๐(๐ + 1)(๐ + 2))/3 We will prove that P(k + 1) is true. 1.2 + 2.3 + 3.4 +โฆ.+ (k + 1).((k + 1) + 1) = ((๐ + 1)((๐ + 1)+ 1)((๐ + 1)+ 2))/3 1.2 + 2.3 + 3.4 +โฆ.+ (k + 1).(k + 2) = (๐ + 1)(๐ +2)(๐ +3)/3 1.2 + 2.3 + 3.4 +โฆ. + k.(k + 1) + (k + 1).(k + 2) = (๐ + 1)(๐ +2)(๐ +3)/3 We have to prove P(k+1) from P(k) From (1) 1.2 + 2.3 + 3.4 +โฆ.+ k.(k + 1) = (๐(๐ + 1)(๐ + 2))/3 Adding (k + 1). (k + 2) both sides 1.2 + 2.3 + 3.4 +โฆ.+ k.(k + 1) + (k + 1).(k + 2) = (๐(๐ + 1)(๐ + 2))/3 + (k + 1).(k + 2) = (๐(๐ + 1)(๐ + 2)+3(๐ + 1)(๐ + 2))/3 = ((๐ + 1)(๐ + 2) (๐ + 3) )/3 Thus, 1.2 + 2.3 + 3.4 +โฆ. + k.(k + 1) + (k + 1).(k + 2) = (๐ + 1)(๐ +2)(๐ +3)/3 which is the same as P(k + 1) 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

Ex 4.1