Mathematical Proof/Appendix/Answer Key/Mathematical Proof/Methods of Proof/Counterexamples

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(Q).Using the method of induction,show that for all n € N,


Soln: Let us take

S(n)= 1+2+3+....+n=n(n+1)/2,n € N

When,n=1,then , L.H.S.=n , R.H.S.=n(n+1)/2

     =1 ,        =1(1+1)/2

.°. L.H.S.=R.H.S,so,the statement given is true for n=1

Let,the statement be true for n=k i.e.S(k)=>1+2+3+....+k=k(k+1)/2(i)

Again,Let the statement be true for n=k+1 i.e=>1+2+3+....+k+(k+1)=(k+1)(k+1+1)/2

Now,adding k+1 to both the sides of eqn (i).



Hence,the statement given for n=k+l is also true.

So,it is true for all n € N