Prove that 1 + 3n ≤ 4n , for every integer n ≥ 0.
Where did you get stuck? This one was not that difficult. I'll just give the solution. It doesn't work for zero, you can check this.
Define a statement by for all integers
We show that is true using mathematical induction.
Since , we have that is true.
Assume that is true, we show that holds also.
Note that,
So that , which is . Thus, holds by induction.