DrDelMath

SUPPLEMENT
Mathematical Induction

Principle of Mathematical Induction
Let there be associated with each natural number n a proposition P_n which is either true or false.
     If   P₁ is true   and    P_k implies P_k+1
     then P_n is true for all natural numbers

Remark: Simply "testing" a proposition for many cases is not sufficient to conclude that the proposition will be true for all cases.
As an illustration consider the following proposition:
    P_n is the statement that n² -79n +1601 is a prime.

P_n is the statement that 1 - 79 + 1601 = 1523 is a prime; which is true.

P₂ is the statement that 2² - (79)(2) + 1601 = 1447 is a prime; which is true.

P₃ is the statement that 3² - (79)(3) +1601 = 1373 is a prime; which is true.

           continue testing P_n     all will be true

P₇₉ is the statement that 79² -(79)(79) + 1601 = 1601 is a prime; which is true.

HOWEVER    P₈₀ is the statement that 80² -(80)(79) + 1601 = 1681 is a prime; which is false because 1681 = (41)(41)

Remark: To prove that P_k implies P_k+1 is not sufficient to conclude that the proposition is true for all natural numbers.
It is necessary to prove that P₁ is true.
To illustrate this point consider the following;