Show that 9ⁿ⁺¹ - 8n - 9 is divisible by 64, whenever n is a positive integer.

n = 1 ⇒ 9^{n + 1} - 8n - 9 = 9² - 8 - 9
                                = 81 - 17 = 64= 1(64)
n = 2 ⇒9^{n + 1} - 8n - 9 = 9³ - 8(2) - 9
                                = 729 – 16 - 9 = 704= 11 (64)
From n = 3, 4, 5,.....9^{n + 1} – 8n - 9 = 9(1 + 8)n - 8n - 9
                                                 = 9 [ⁿC₀ + ⁿC₁ . 8 + ⁿC₂.8² + ... ⁿC_n 8ⁿ] – 8n - 9
                                                 = 9[1 + 8n + ⁿC₂.8² + ... ⁿC_n 8n] –8n – 9
                                                 = 9 + 72n + 9. ⁿC₂. 8² + ... 9 ⁿC_n 8ⁿ –8n - 9
                                                 = 8² [n + 9 (ⁿC₂ + ⁿC₃.8 +... ⁿC_n 8^n-2)]
                                                           which is divisible by 64.