Abstract
With the algebraic background of the previous chapters at our disposal, we devote the first section of this chapter to the development of a very important topic in elementary Mathematics, namely, the principle of mathematical induction. It will considerably improve our ability of elaborating proofs of mathematical statements that depend on natural numbers. In the other two sections, we shall apply this principle to deduce Newton’s formula for binomial expansion, as well as some important related results.
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Notes
- 1.
More generally, it can be shown that F n is a perfect square if and only if n = 1 or 12. For a proof of this fact, see the problems of Section 12.3 of [5].
- 2.
After Édouard Zeckendorf , belgian mathematician of the XX century.
- 3.
Another proof is the object of Problem 8.
- 4.
After Michael Stifel , german mathematician of the XVI century.
- 5.
After Blaise Pascal , french mathematician of the XVII century. Besides the triangle that bears his name, there is an important Pascal’s theorem in the theory of conics, which has greatly motivated the developments of Projective and Algebraic Geometry.
- 6.
Sir Isaac Newton , english mathematician and physicist of the XVII century, is considered to be one of the greatest scientists ever. Actually, it is difficult to properly address Newton’s contribution to the development of science. Known as the father of modern Physics, Newton also created, together with G. W. Leibniz, the Differential and Integral Calculus. His masterpiece, Philosophiae Naturalis Principia Mathematica, is one of the most influential books ever written and contains the cornerstones of both Calculus and Physics.
- 7.
Joseph Louis Lagrange , french physicist and mathematician of the XVIII century. Lagrange was one of the greatest scientists of his time, with notable contributions to Physics and Mathematics. In particular, he was a pioneer in the fields of Calculus of Variations and Celestial Mechanics.
Bibliography
A. Caminha, An Excursion Through Elementary Mathematics III - Discrete Mathematics and Polynomial Algebra (Springer, New York, 2018)
E. Scheinerman, Mathematics, a Discrete Introduction (Cengage Learning, Boston, 2012)
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Caminha Muniz Neto, A. (2017). Induction and the Binomial Formula. In: An Excursion through Elementary Mathematics, Volume I. Problem Books in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-53871-6_4
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DOI: https://doi.org/10.1007/978-3-319-53871-6_4
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