1. The problem statement, all variables and given/known data Question: Use induction to prove that 3^n > n x 2^n for every natural number n ≥ 3 2. Relevant equations N/A 3. The attempt at a solution Answer: Step 1: 3^3 > 3 x 2^3 ⇒ 27 > 24 Step 2: Assume 3^k > k x 2^k Step 3: 3^(k+1) > (k+1) x 2^(k+1) ⇒ 3 x 3^k > k x 2^(k+1) + 2^(k+1) ⇒ 3^k + 3^k + 3^k > k x 2^k + k x 2^k + 2^k + 2^k. So am I allowed to use the assumption as a given to make the green and red parts of the inequality true? And am allowed to just plug in the base k value that would show the black parts of the inequality to be true? Another thing that I don't understand about induction is why we have to assume that the inequality in the question is true for k. If k can only be the base case in the beginning, then why do we have to assume that it works for some k when we just showed that it works when k is the base case?