Let P(n) be the proposition that 7n−1 is divisible by 6.

Verify P(1):

which is divisible by 6. Hence, P(1) is true.

Assume P(k) is true:

Show that P(k+1) is true: 

Consider

By the induction hypothesis, 7k−1=6m, so:     

which is divisible by 6. Thus, P(k+1) is true.

Conclusion

We have shown that P(1) is true, and that if P(k) is true then P(k+1) is true. Hence, by induction,   7n−1 is divisible by 6 for all n≥1.