Question

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

Answer 1

9ⁿ⁺¹ = (1+8)ⁿ⁺¹ = ⁿ⁺¹C₀ + ⁿ⁺¹C₁(8) + ⁿ⁺¹C₂(8)² +.....+ ⁿ⁺¹C_n+1(8)ⁿ⁺¹

= 1+(n+1)(8) + 8² [ⁿ⁺¹C₂ + ⁿ⁺¹C₃ x 8 +.....+ ⁿ⁺¹C_n+1(8)ⁿ⁻¹]

= 9 + 8n + 64[ⁿ⁺¹C₂ + ⁿ⁺¹C₃ x 8 +....+ ⁿ⁺¹C_n+1(8)ⁿ⁻¹]

⇒ 9ⁿ⁺¹−8n − 9 = 64k,

Where k = ⁿ⁺¹C₂ + ⁿ⁺¹C₃x8 +....+ ⁿ⁺¹C_n+1(8)ⁿ⁻¹ which is a natural number

Thus, 9ⁿ⁺¹ − 8n − 9 is divisible by 64, whenever n is a positive integer.