3²ⁿ⁺² – 8n – 9 is divisible by 8.

Let P(n) = 3²ⁿ⁺² – 8n – 9
P(1) is divisible by 8
Let us assume P(k) is divisible by 8.
P(k) = 3^2k+2 – 8k – 9 = 8M………….. (A)
To prove P(k+1) is divisible by 8 using the result of (A)
P(k+1) = 3^2(k+1)+2 – 8(k+1) – 9
          = 3^2k+2+2 – 8k – 8 –9
          = 3^2k+2 . 3² – 8k- 9 –8
          = (8M + 8K + 9) 9 – 8k – 9 – 8
          = 72 M + 72 K + 81 – 8K – 9 – 8
          = 72 M + 64 k +64
          = 8(9M + 8K + 8)
P( K + 1) is divisible by 8.
hence the result.
P(K+1) is true.
By the Principle of mathematical induction, P(n) is true for all values of nwhere n N
Hence proved