Assignment I

Name : Isnaini Nurul Noviana
NIM : 07305141025

CHAPTER I

INDUCTION OF MATHEMATICS AND BINOMIAL THEOREM

A. Induction of mathematics

Induction of mathematics is one of the proof method from many theorems in theory of number or in other mathematics. Whereas binomial theorem, besides as basic it used to derivation some binomial theorem and solution of mathematics problem. Because of that, the capable of this skill is necessary to who will learn mathematics, because there many treatises in mathematics which use that principal to derivate the theorem or to solve problem. Most of each next treatise use this both principal, either to prove the theorem or to solve the problems.

Induction of mathematics is one of the proof argumentation of a theorem or mathematics statement which whole object is set of integer number or especially set of original number. Pay attention to example mathematics statements bellow.

Example 1.1
1+2+3+…+n= ½ n(n+1) for each original number of n. Is this statement true? For replied this question, we can try to substitute n on this statement into any original number. If n = 1, then that statement become 1= ½ .1(1+1), or 1 = 1, that is got a true statement. If n = 2, then that statement become 1+2= ½ .2(2+1), , or 3 = 3, that is got a true statement. If n = 3, then that statement become 1+2+3= ½ .3(3+1), , or 6 = 6, that is got a true statement too.

The reader continueable to n = 4; 5; or other original number and usually will get true statement. Is with give some example with substitute some of original number into n from origin statement and got true statements can give proof about truth of this statement?

In mathematics, gift some example like that, is not proof of truth of statement which can used in the whole set. The statement in example above, the whole of set is set of all of original number. If we give an example for each original number of n in this statement and get true statement for each number, then that is as truth of proof from that statements.

But this is not efisien and imposible to do, because the element of set of original number unfinite. So, how to prove that statement? One of the way is view the first part from this statement as arithmetics series with first component a = 1, the difference b = 1, final component is Un= n and has n component. So the sum of this series is
Sn= ½ n(a+Un)
= ½ n(1+n)
= ½ n(n+1), that is second part from proved statement.