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