The Origin of

Mathematical Induction. The American Mathematical Monthly, 24(5), 199-207.

Both the proof by direct calculation of determinant in Section 3 and the proof by

mathematical induction method in Section 4 lead to the same result; namely, the determinant [absolute value of A - [lambda]E] equal to [(-[lambda]).sup.n-1](n - [lambda]) for arbitrary consistent matrix A with arbitrary natural n (n [greater than or equal to] 2).

Utilizing (3.44) and by

mathematical induction on [alpha], we obtain

Based on the idea of the

mathematical induction method, we have

Two examples of proof by

mathematical induction are suggested: the first would illustrate "sums", and the other "divisibility results".

Then, we find that [absolute of (c)] has the lowest modulus than any other points with modulus r in the following by using

mathematical induction.

Keywords Sets of n-odd prime numbers, Pairs of consecutive odd prime numbers,

Mathematical induction, Odd points, Positive directional half line of the number axis, [RLSS.sub.No1~NoX], Sets of *[mu](*s)+b([??]s)*, Pairs of *v([??]s)*, The coexisting theorem, No1 [RLS.sub.No1~NoX], Set of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], Pair of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

On the other hand, using the recursive formulas above, we can prove the second equation for j < l or [j.sup.*] < l, and the third equation for j + 1 < l or [j.sup.*] + 1 < l simultaneously by double

mathematical induction on l, and k + [k.sup.*], that is, the main induction on l and the supplementary induction on k + [k.sup.*].

Hence, by virtue of the

mathematical induction, we have [I'.sub.n](r) < 0 for all n [member of] N and 0 [less than or equal to] r < 1.

Instead, it is shown in Appendix A with

mathematical induction [18,19] that an approximation of [SINR.sub.L] for any 2 [less than or equal to] q < p is given by

Now, relation (5) follows by

mathematical induction, the induction step [y.sub.1] ...

By using

mathematical induction, the general solution of a polyanalytic differential equation of the Fempl type is