Question

Prove the following by using the principle of mathematical induction for all n ∈ N :

Answer 1

The given statement is,

P( n ):( 1+ 1 1 )( 1+ 1 2 )( 1+ 1 3 )...( 1+ 1 n )=( n+1 )(1)

For n=1,

P( 1 ):( 1+ 1 1 )=( 1+1 ) P( 1 ):2=2

Thus, P( 1 ) is true.

Substitute n=k in equation (1).

P( k ):( 1+ 1 1 )( 1+ 1 2 )( 1+ 1 3 )...( 1+ 1 k )=( k+1 )(2)

According to the principle of mathematical induction, assume that the statement P( n ) is true for n=k. If it is also true for n=k+1, then P( n ) is true for all natural numbers.

Substitute n=k+1 in equation (1).

P( k+1 ):( 1+ 1 1 )( 1+ 1 2 )( 1+ 1 3 )...( 1+ 1 k )( 1+ 1 k+1 )=( k+1+1 )(3)

Substitute the value from equation (2) in equation (3).

P( k+1 ):( k+1 )( 1+ 1 k+1 )=( k+2 ) P( k+1 ):( k+1 )( k+1+1 k+1 )=( k+2 ) P( k+1 ):( k+2 )=( k+2 )

It is proved that P( k+1 ) is true whenever P( k ) is true.

Hence, statement P( n ) is true.