Prove the following by using the principle of mathematical induction for all *n* ∈ *N*: 3^{2} ^{ n } ^{ + 2} – 8*n* – 9 is divisible by 8.

Let the
given statement be P(*n*), i.e.,

P(*n*):
3^{2}^{n}^{ + 2} – 8*n* –
9 is divisible by 8.

It can be
observed that P(*n*) is true for *n* = 1 since 3^{2 }^{×}^{
1 + 2} – 8 × 1 –
9 = 64, which is divisible by 8.

Let P(*k*)
be true for some positive integer *k*, i.e.,

3^{2}^{k}^{
+ 2} – 8*k* – 9 is divisible by 8.

∴3^{2}^{k}^{
+ 2} – 8*k* – 9 = 8*m*; where *m* ∈
**N** … (1)

We shall
now prove that P(*k* + 1) is true whenever P(*k*) is true.

Consider

Thus, P(*k*
+ 1) is true whenever P(*k*) is true.

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

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