StrongInduction.md (1004B)
1 # Strong Induction 2 3 Abstract Math 10.2. Weak induction is the normal form of induction discussed in [Induction](Induction.md) 4 5 **Definition:** Strong induction is the process of proving one or more prior true statements implies a later one much like weak induction, but with strong induction we can prove in the form of $S_{k-5} \implies S_{k+1}$ so long as k-5 is in the domain and that every value between k-5 and k+1 has been shown to be true. 6 7 Steps: 8 9 1. Prove the first statement $S_1$ or more if needed. 10 2. Given any integer k$\geq$ 1, prove $(S_1 \wedge S_2 \wedge S_3 \wedge ... \wedge S_k) \implies S_{k+1}$ 11 12 A good example of this is an equation that does not factor nicely. If I know that $S_1$ is true, but I can't factor $S_2$ in a satisfactory way to prove that for each n+1 the statement is true, then proving a few until finding an instance of something factoring well can solve this issue. 13 14 Can be used to prove [Fundamental Theorem Of Arithmetic](FundamentalTheoremOfArithmetic.md)