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Question 26478: Use the principle of mathematical induction to prove the following identity.
If n is greator and equal to 1 then,
1*2+.......+n(n+1) = (n(n+1)(n+2))/3
Answer by venugopalramana(3286)   (Show Source):
You can put this solution on YOUR website! 1*2+.......+n(n+1) = (n(n+1)(n+2))/3
IN INDUCTION ,WE FIRST TEST WHETHER IT IS CORRECT FOR N=1
LHS=1*2=2
RHS=1*2*3/3=2...LHS=RHS...SO IT IS TRUE FOR N=1
NOW WE ASSUME IT IS TRUE FOR N=K ......SOME VALUE...SO
1*2+.......+K(K+1) = (K(K+1)(K+2))/3........................I
WE NOW GO ON TO PROVE THAT IT IS TRUE FOR N=K+1...THAT IS...TPT
1*2+.......+K(K+1)+(K+1)(K+2) = ((K+1)(K+2)(K+3))/3..........II
LHS={1*2+.......+K(K+1)}+(K+1)(K+2)...USING EQN.I
=(K(K+1)(K+2))/3}+(K+1)(K+2)
=(K+1)(K+2){(K/3)+1}
=(K+1)(K+2){(K+3)/3}
=((K+1)(K+2)(K+3))/3=RHS OF EQN.II.
SO WE GOT IT NOW THAT ,IF THE GIVEN RELATION IS TRUE FOR N=K,THEN IT IS TRUE FOR N=K+1
BUT WE PROVED FIRST THAT IT IS TRUE FOR N=1
SO IT IS TRUE FOR N=2
SO IT IS TRUE FOR N=3
SO IT IS TRUE FOR N=4
........ETC..........
SO IT IS TRUE FOR ALL INTEGRAL VALUES OF N
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