Check sibling questions

Ex 4.1, 2 - 13 + 23 + 33 + .. + n3 = (n(n + 1)/2)2 by induction - Ex 4.1

Ex 4.1, 2 - Chapter 4 Class 11 Mathematical Induction - Part 2
   Ex 4.1, 2 - Chapter 4 Class 11 Mathematical Induction - Part 3


Transcript

Ex 4.1,2: Prove the following by using the principle of mathematical induction 13 + 23 + 33+ + n3 = ( ( +1)/2)^2 Let P (n) : 13 + 23 + 33 + 43 + ..+ n3 = ( ( +1)/2)^2 For n = 1, L.H.S = 13 = 1 R.H.S = (1(1 + 1)/2)^2= ((1 2)/2)^2= (1)2 = 1 Hence, L.H.S. = R.H.S P(n) is true for n = 1 Assume that P(k) is true 13 + 23 + 33 + 43 + ..+ k3 = ( ( + 1)/2)^2 We will prove that P(k + 1) is true. 13 + 23 + 33 + ..+ k3 + (k + 1)3= ((k + 1)((k + 1)+ 1)/2)^2 13 + 23 + 33 + ..+ k3 + (k + 1)3= ((k + 1)( + 2)/2)^2 We have to prove P(k+1) from P(k) i.e. (2) from (1) From (1) 13 + 23 + 33 + 43 + ..+ k3 = ( ( + 1)/2)^2 Adding (k+1)3 both sides, 13 + 23 + 33 + ..+ k3 + (k + 1)3 = ( ( + 1)/2)^2+ (k + 1)3 = ( ^2 ( + 1)^2)/2^2 + (k + 1)3 = ( ^2 ( + 1)^2)/4 + (k + 1)3 = (k^2 (k + 1)^2 + 4( + 1)^3)/4 = ((k + 1)^2 (k^2 + 4 + 4))/4 = ((k + 1)^2 (k^2 + 2 + 2 + 4))/4 = ((k + 1)^2 ( ( + 2) + 2( + 2)))/4 = ((k + 1)^2 (( + 2) ( + 2)))/4 = ((k + 1)^2 (k + 2)^2)/4 = ((k + 1)( + 2)/2)^2 Thus, 13 + 23 + 33 + ..+ k3 + (k + 1)3= ((k + 1)( + 2)/2)^2 i.e. 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

Davneet Singh's photo - Co-founder, Teachoo

Made by

Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.