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

(Q).Using the method of induction,show that for all n € N,   1+2+3+....+n=n(n+1)/2 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                  =1×2/2                  =2/2                  =1 .°. 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 =(k+1)(k+2)/2(ii) Now,adding k+1 to both the sides of eqn (i).

.i.e.=>1+2+3+....+k+(k+1)=k(k+1)/2+(k+1) =(k+1){k/2+1} =(k+1){k+2/2} =(k+1)(k+2)/2(iii) .°.(iii)=(i)

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

So,it is true for all n € N                        #solved#