Question

mathematical induction: prove that 8 is a factor of 9^n-1 for all positive integers n

Answer 1

You can actually do this without induction, is it required to do it by induction?

Answer 2

for n=1

9-1 = 8

hence true for integer n...so 9^n - 1 = 8p

for (n+1)

9^(n+1) - 1 = 9.9^n -1 - 8 + 8 = 9.9^n - 9 + 8

= 9(9^n - 1) + 8 = 9.8p + 8 = 8(9p+1)

Answer 3

understand?

Answer 4

\[9^{(n+1)}-1 = 9^n + 9^n ...9 \times - 1 = 9^n -1 + 9^n -1 .... 9 \times + 8\] each one of the individual terms are already divisible by 8 using the induction hypothesis the last term also is clealty divisible by 8