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from IV - Techniques of proof
Simplicity is the ultimate sophistication.
Leonardo da VinciIn this chapter we investigate more sophisticated versions of induction. There are three variants we shall be most interested in.
(i) We use a different initial case. Rather than show that A(1) is true we show, for instance, A(7) or A(15) is true. Thus A(n) is true for all n ≥ 7 or all n ≥ 15 respectively.
(ii) We change the inductive step to ‘A(k − 1) and A(k) imply A(k + 1).’ This requires us to have as initial case that A(1) and A(2) are true.
(iii) We change the inductive step to ‘A(j) true for all 1 ≤ j ≤ k implies A(k + 1) true.’ We use initial case A(1) true or some other initial case like (i) above.
All three can be referred to as the Principle of Mathematical Induction and, in addition, the latter two are sometimes called the Principle of Strong Mathematical Induction.
First variant
We do not need to start with n = 1 as the initial case. For example, for statements A(n) the first few cases may be false. If we can show
(i) A(r) is true for some r ∈ ℕ, and
(ii) A(k) ⇒ A(k + 1) for all k ≥ r,
then the statement is true for all A(n) with n ≥ r. Observe that our main change to induction is really the initial step – the change of the range of values in the inductive step is minimal.
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