Use mathematical induction to prove that if n people stand i

Use mathematical induction to prove that if n people stand in a line, where n is a positive integer, and if the first person in the line is right-handed and the last person in fine is left-handed, then somewhere in the line, there is a right-handed person directly in front of a left-handed person.

Solution

There are n persons in the queue with first one right handed and last one left handed.

Let n=2.Then there are only two persons with first one right and second left. Hence there is a right handed person directly in front of left handed

True for n =2.

Assume true for n =k

To prove true for n =k+1

Since true for n =k for k persons there is a right handed person say mth one directly in front of m+1th (left handed person)

If we add one person below m and above m+1, then again there is a right handed person say mth one directly in front of m+1th (left handed person).

Instead if we insert the k+1th person in between mth and m+1th persons, then the new person if right handed would be directly in front of m+1th left handed

Or if the new person is left handed then mth person is the right handed who would be directly in front of new person left handed.

Thus true for k+1 if true for k

Already true for k =2

Hence by Mathematical induction true for all natural numbers n.