1. The problem statement, all variables and given/known data Use proof by induction to show that 2^n > n^3 for n>9 2. Relevant equations 3. The attempt at a solution My solution is not as required by the question because I cannot really understand the proof by induction. I will give summary of my understanding. Please check my understanding. Basically we use the first equation ## 2^n > n^3 ## to check if it is true for n = 10. If it is true then we check for n = 11 and we can go on checking to infinity. To cut this indefinitely long process, we put ##n = n+1## in place of n. so that it becomes ## 2^(n+1) > (n+1)^3##. Why do we do this? Is it because the difference between any consecutive integer is 1 so we check n+1 the next integer. If we have a formula that is true for n and the next integer. i.e. ## 2^n > n^3 ## and ## 2^(n+1) > (n+1)^3## but we want to prove that they are true in the first place. We can start from n =1 and n = n+1 = 2 then the next n = 2 and n = n + 1 =3 and so on to infinity. Each n+1 will take the place of n and so on ? From experience I can tell that in equations where proving "something" = "somethingelse" for all n. I try to prove the customary step "something + 1" = "somethingelse + 1" with some substitution from "something" = "somethingelse". if the LHS and RHS equal then it is true for all n. I have no understanding of why it is done like this? Anyway my method was that to draw two graphs ##y = 2^n## and ##y = (n+1)^3## on the same plot and show that the gradient of one cure is more positive than the gradient of another curve after n > 9? I have shown the plot below. Is this acceptable? Danke..