How do I prove by induction that n^2 is greater than or equal to 3n + 4 for n > 3?
The simplest and most common form of mathematical induction proves that a statement involving a natural number n holds for all values of n. The proof consists of two steps:
- The basis (base case): showing that the statement holds when n is equal to the lowest value that n is given in the question. Usually, n = 0 or n = 1.
- The inductive step: showing that if the statement holds for some n, then the statement also holds when n + 1 is substituted for n.
The assumption in the inductive step that the statement holds for some n is called the induction hypothesis (or inductive hypothesis). To perform the inductive step, one assumes the induction hypothesis and then uses this assumption to prove the statement for n + 1.