Answer:
Step-by-step explanation:
1) first of all, let s check for n = 1
[tex]7^1 -1=7-1=6[/tex]
that s true
2) We assume that this is true for n
[tex]7^n-1[/tex] is divisible by 6
what about [tex]7^{n+1}-1[/tex] ?
we know that there is a k natural so that [tex]7^n-1=6k[/tex]
so [tex]7^n = 1+6k[/tex]
then [tex]7^{n+1} = 7*7^n = 7(1+6k)\\[/tex]
so [tex]7^{n+1}-1 = 7(1+6k)-1 = 6+7*6k = 6(1+7k)[/tex]
so it means that [tex]7^{n+1}-1[/tex] is divisible by 6
3) finally as this is true for n=1 and if this is true for n then it is true for n+1 we can conclude that [tex]7^n-1[/tex] is divisible by 6 for n positive integer
