web analytics

Ex 4.1, 4 - Prove 1.2.3 + 2.3.4 + .. + n(n + 1) (n + 2) = n(n+1) - Equal - Addition

  1. Chapter 4 Class 11 Mathematical Induction
  2. Serial order wise
Ask Download

Transcript

Ex 4.1, 4: Prove the following by using the principle of mathematical induction for all n โˆˆ N: 1.2.3 + 2.3.4 + โ€ฆ + n(n + 1) (n + 2) = (๐‘›(๐‘› + 1)(๐‘› + 2)(๐‘› + 3))/4 Let P(n): 1.2.3 + 2.3.4 + โ€ฆ + n(n + 1) (n + 2) = (๐‘›(๐‘› + 1)(๐‘› + 2)(๐‘› + 3))/4 For n = 1, L.H.S = 1.2.3 = 6 R.H.S = (1(1+1)(1+2)(1+3))/4 = (1 ร— 2 ร— 3 ร— 4)/4 = 6 L.H.S. = R.H.S โˆดP(n) is true for n = 1 Assume P(k) is true 1.2.3 + 2.3.4 + โ€ฆ + k(k + 1) (k + 2) = (๐‘˜(๐‘˜ + 1)(๐‘˜ + 2)(๐‘˜ + 3))/4 We will prove that P(k + 1) is true. 1.2.3 + 2.3.4 + โ€ฆ + (k + 1) ((k +1) + 1) ((k + 1) + 2) = (๐‘˜ + 1)((๐‘˜ + 1)+ 1)((๐‘˜ + 1)+ 2)((๐‘˜ + 1)+ 3)/4 1.2.3 + 2.3.4 + โ€ฆ + (k + 1) (k + 2) (k + 3)ย  = ((๐‘˜ + 1)(๐‘˜ + 2)(๐‘˜ + 3)(๐‘˜ + 4))/4 1.2.3 + 2.3.4 + โ€ฆ + k(k + 1) (k + 2) +(k + 1) (k + 2) (k + 3) = ((๐‘˜ + 1)(๐‘˜ + 2)(๐‘˜ + 3)(๐‘˜ + 4))/4 We have to prove P(k+1) from P(k) i.e. (2) from (1) From (1) 1.2.3 + 2.3.4 + โ€ฆ + k(k + 1) (k + 2) = (๐‘˜(๐‘˜ + 1)(๐‘˜ + 2)(๐‘˜ + 3))/4 Adding (k+1) (k+2) (k+3) both sides 1.2.3 + 2.3.4 + โ€ฆ + k(k + 1) (k + 2) +(k + 1) (k + 2) (k + 3) = (๐‘˜(๐‘˜ + 1)(๐‘˜ + 2)(๐‘˜ + 3))/4 +(k + 1) (k + 2) (k + 3) = (๐‘˜(๐‘˜ + 1)(๐‘˜ + 2)(๐‘˜ + 3) + 4(๐‘˜ + 1)(๐‘˜ + 2)(๐‘˜ + 3))/4 = ((๐‘˜ + 1)(๐‘˜ + 2)(๐‘˜ + 3) (๐‘˜ + 4))/4 Thus, 1.2.3 + 2.3.4 + โ€ฆ + k(k + 1) (k + 2) +(k + 1) (k + 2) (k + 3) = ((๐‘˜ + 1)(๐‘˜ + 2)(๐‘˜ + 3)(๐‘˜ + 4))/4 which is the same as P(k + 1) โˆด P(k + 1) is true whenever P(k) is true. โˆด By the principle of mathematical induction, P(n) is true for n, where n is a natural number

About the Author

Davneet Singh's photo - Teacher, Computer Engineer, Marketer
Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He provides courses for Mathematics from Class 9 to 12. You can ask questions here.
Jail