Assignment #3: Mathematical Induction

Transcription

1 Math 3AH, Fall 011 Section Assignment #3: Mathematical Inuction Directions: This assignment is ue no later than Monay, November 8, 011, at the beginning of class. Late assignments will not be grae. You will be grae on exactly what is ase for in the instructions below. You shoul submit your wor on a separate sheet of paper in the orer the questions are ase. You will not only be grae on your mathematics, but also on your organization, proper use of English, spelling, punctuation, an logic. Purpose: In this assignment you will stuy a form of mathematical proof calle mathematical inuction. In general, inuction is use to prove statements that involve an integer n in their statement. Examples of such statements are: The sum of the first n integers: The sum of all o integers up to a certain integer: Integrating the sum of n functions: Introuction: The simplest an most common form of mathematical inuction proves that a statement involving a natural number n hols for all values of n. The proof consists of two steps: The Base Case: Here we show that the statement hols true when n is equal to the lowest possible value that n can be given in the question. Usually that s n 0 or n 1. The Inuctive Step: Here we show that if the statement hols for some arbitrary value of n, then the statement also hols when n1is substitute for n. The assumption in the inuctive step that the statement hols for some arbitrary n is calle the inuction hypothesis (or inuctive hypothesis). To perform the inuctive step, one assumes the inuction hypothesis an then uses this assumption to prove the statement for n 1. This metho wors by first proving the statement is true for a starting value, an then proving that the process use to go from one value to the next is vali. If these are both proven, then any value can be obtaine by performing the process

2 Math 3AH, Fall 011 Section repeately. It may be helpful to thin of the omino effect; if one is presente with a long row of ominoes staning on en, one can be sure that: The first omino will fall Whenever a omino falls, its next neighbor will also fall, so it is conclue that all of the ominoes will fall, an that this fact is inevitable. Example 1: One application of mathematical inuction is to prove a variety of sum formulas. For example, the sum of the whole numbers starting with 1 up to some arbitrary integer n obeys the following pattern n n n This pattern motivates the following proof by inuction. [Proof] I will prove that the sum of the first n positive integers can be given by the formula nn n. 1 nn1 Base Case: Lets show it s true for n 1. Above we saw that 1, an thus our formula hols for the first possible value of n. Inuctive Step: Let s suppose our statement is true for an arbitrary value n an let s show it s true for n 1. That is, we want to show that

3 Math 3AH, Fall 011 Section With a little algebraic manipulation we see that Therefore our statement is true for n 1. Since the basis an inuctive step have been prove, our statement is true by mathematical inuction. Example : Another example of a sum formula epenent on an integer n is that of the sum of the first n os. The pattern is as follows: n1 n This pattern motivates the following proof by inuction. [Proof] I will prove that the sum of the first n o integers can be given by the formula n1 n Base Case: Let s show it s true for n n Inuctive Step: Let s suppose our statement is true for an arbitrary value n an let s show it s true for n 1. That is, we want to show that

4 Math 3AH, Fall 011 Section Therefore our statement is true for n 1. Since the basis an inuctive step have been prove, our statement is true by mathematical inuction. Now it s time for you to try some! 1. Prove the following formulas by mathematical inuction. a) b) n n n n n n n Example 3: Another application of mathematical inuction is to exten properties about two objects to properties about n objects. For example, if f an g are both functions ifferentiable at x, then the function f g is ifferentiable at x an f x g x f x g x. However, it seems reasonable that ifferentiation shoul istribute over aition no matter how many functions are being ae. That is, n j1 Here s a proof of this fact using inuction. f j x f1 x fnx f1 x fn x n f j x j1

5 Math 3AH, Fall 011 Section [Proof] Base Case: The fact that we nee at least two functions to a implies we shoul start with n. However this is trivial since this is the property about ifferentiation that motivate us in the first place: f1 x f x f1 x f x Inuctive Step: Let s suppose our statement is true for an arbitrary value n an let s show it s true for n 1. That is let s assume that f jx f jx () j1 j1 This isn t so ba! Let s just apply our Base Case nowlege an our assumption. 1 f x f x f x j j 1 j1 j1 j1 j1, f j x f 1 x, j 1 f j x f 1 x, 1 f j x. by our Base Case by () Therefore our statement is true for n 1. Since the basis an inuctive step have been prove, our statement is true by mathematical inuction. Now it s time for you to try some!. Prove the following statements about functions using mathematical inuction. a) Suppose f1, f,, fn are all continuous functions an c1, c,..., cn are constants. Then the function n f x c f x is also continuous. Here you shoul use facts proven in assignment #. 1

6 Math 3AH, Fall 011 Section b) Suppose f1, f,, fn are all integrable functions. Then their sum is also integrable an c) Generalize Prouct Rule: Suppose f1, f,, fn are all ifferentiable functions at x. Then their prouct is also ifferentiable at x an f1 x f x fnx f1 x f x fnx f1 x f x fn x f1 x f x fn x