**Ex 4.1, 6**

Last updated at Dec. 8, 2016 by Teachoo

Last updated at Dec. 8, 2016 by Teachoo

Transcript

Ex 4.1,6: Prove the following by using the principle of mathematical induction for all n β N: 1.2 + 2.3 + 3.4 +β¦β¦ + n. (n + 1) = (π(π + 1)(π + 2))/3 Let P(n): 1.2 + 2.3 + 3.4 +β¦.+ n.(n + 1) = (π(π + 1)(π + 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 β΄ P(n) is true for n = 1 Assume P(k) be true 1.2 + 2.3 + 3.4 +β¦.+ k.(k + 1) = (π(π + 1)(π + 2))/3 We will prove that P(k + 1) is true. 1.2 + 2.3 + 3.4 +β¦.+ (k + 1).((k + 1) + 1) = ((π + 1)((π + 1)+ 1)((π + 1)+ 2))/3 1.2 + 2.3 + 3.4 +β¦.+ (k + 1).(k + 2) = (π + 1)(π +2)(π +3)/3 1.2 + 2.3 + 3.4 +β¦. + 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 +β¦.+ k.(k + 1) = (π(π + 1)(π + 2))/3 Adding (k + 1). (k + 2) both sides 1.2 + 2.3 + 3.4 +β¦.+ 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 +β¦. + k.(k + 1) + (k + 1).(k + 2) = (π + 1)(π +2)(π +3)/3 which is the same as P(k + 1) β΄ P(k + 1) is true whenever P(k) is true. β΄ By the principle of mathematical induction, P(n) is true for n, where n is a natural number

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