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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,
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#