<img height="1" width="1" style="display:none" src="https://www.facebook.com/tr?id=539359806247306&ev=PageView&noscript=1"/>

Ex 4.1, 6 - 1.2 + 2.3 + 3.4 + .. + n.(n+1) = n(n+1)(n+2)/3 - Equal - Addition

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

Transcript

Ex 4.1,6: Prove the following by using the principle of mathematical induction for all n ∈ N: 1.2 + 2.3 + 3.4 +…… + n. (n + 1) = (𝑛(𝑛 + 1)(𝑛 + 2))/3 Let P(n): 1.2 + 2.3 + 3.4 +….+ n.(n + 1) = (𝑛(𝑛 + 1)(𝑛 + 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 ∴ P(n) is true for n = 1 Assume P(k) be true 1.2 + 2.3 + 3.4 +….+ k.(k + 1) = (π‘˜(π‘˜ + 1)(π‘˜ + 2))/3 We will prove that P(k + 1) is true. 1.2 + 2.3 + 3.4 +….+ (k + 1).((k + 1) + 1) = ((π‘˜ + 1)((π‘˜ + 1)+ 1)((π‘˜ + 1)+ 2))/3 1.2 + 2.3 + 3.4 +….+ (k + 1).(k + 2) = (π‘˜ + 1)(π‘˜ +2)(π‘˜ +3)/3 1.2 + 2.3 + 3.4 +…. + k.(k + 1) + (k + 1).(k + 2) = (π‘˜ + 1)(π‘˜ +2)(π‘˜ +3)/3 We have to prove P(k+1) from P(k) From (1) 1.2 + 2.3 + 3.4 +….+ k.(k + 1) = (π‘˜(π‘˜ + 1)(π‘˜ + 2))/3 Adding (k + 1). (k + 2) both sides 1.2 + 2.3 + 3.4 +….+ k.(k + 1) + (k + 1).(k + 2) = (π‘˜(π‘˜ + 1)(π‘˜ + 2))/3 + (k + 1).(k + 2) = (π‘˜(π‘˜ + 1)(π‘˜ + 2)+3(π‘˜ + 1)(π‘˜ + 2))/3 = ((π‘˜ + 1)(π‘˜ + 2) (π‘˜ + 3) )/3 Thus, 1.2 + 2.3 + 3.4 +…. + k.(k + 1) + (k + 1).(k + 2) = (π‘˜ + 1)(π‘˜ +2)(π‘˜ +3)/3 which is the same as P(k + 1) ∴ P(k + 1) is true whenever P(k) is true. ∴ 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 contact him here.
Jail