Question

22. 32n+2 - 8n 9 is divisible by 8.

Answer 1

Let the given statement be P( n ) .

P( n ): 3 2n+2 −8n−9 (1)

Here, n is defined for natural numbers.

Substitute n=1 in statement (1).

P( 1 ): 3 2×1+2 −8×1−9=64

64 is divisible by 8 ; hence, P( 1 ) is true.

Let P(k) be true for some positive natural number k. Substitute n=k in statement (1).

P( k ): 3 2k+2 −8k−9

We can write, 3 2k+2 −8k−9=8m where, m∈N (2)

Now we wish to prove that P(k+1)is true whenever P(k) is true.

Substitute n=k+1 in statement (2).

P( k+1 ): 3 2( k+1 )+2 −8( k+1 )−9= 3 2k+2 ⋅ 3 2 −8k−8−9 = 3 2 ( 3 2( k+1 ) −8k−9 )+ 3 2 ( 8k+9 )−8k−17 [ Add and subtract  3 2 ⋅8k and  3 2 ⋅9 ] =9.8m+9( 8k+9 )−8k−17

Further simplify.

P( k+1 ): 3 2( k+1 )+2 −8( k+1 )−9 =9⋅8m+72k+81−8k−17 =9⋅8m+64k+64 =8( 9m+8k+8 )

The above expression is divisible by 8 , hence P( k+1 ) is true when P( k ) is true.

Hence, by the principle of mathematical induction, statement P( n ) is true for all-natural numbers.