Prove that 9n 1 is divisible by 8 for all integers n 0 Ple

Prove that 9^n ? 1 is divisible by 8, for all integers n ? 0.

Please show all steps!

Solution

9ⁿ- 1

= (8+1)ⁿ -1

expanding by binomial thei[orem

nC₀ 1ⁿ + nC₁ 8 + nC₂* 8² + nC₃ 8³ + nC₄ 8⁴ +.................................   to nC_n-1 8^n-1 + nC_n 8ⁿ         - 1

Here each term is a factor of 8 except 1st term which is 1 and last term is -1 . Sothey cancle out. and hence the net number is divsible by 8