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Math 1210-007 Mathematical Proofs Fall 2010 Geometric Series Proof Geometric Series: The sum of the first n powers of a number r, which we shall call S n , can be written concisely in the following way: n S n=∑ k=0 r k =1rr 2...r n= 1−r n1 1−r Proof: By definition we know that 2 S n=1r r r n (1) Multiplying both sides of the equation by r yields rS n =r r 2r n1 (2) Subtracting equation (2) from equation (1) yields the following [Notice all the terms that cancel] S n−rS n=1−r n1 Factoring out the quantity S n on the left hand side gives S n 1−r =1−r n1 Finally, dividing both sides by the quantity 1−r yields the desired equation n1 S n= 1−r 1−r Q.E.D. Math 1210-007 Mathematical Proofs Fall 2010 Proof By Induction Proof by induction is one method of proof for problems where the goal is to demonstrate that a formula works for all natural numbers n. This is done in two steps. Step 1: Show that the formula works when n=1 Step 2: Assume that the formula works for a number n and show that it must be the case that the formula works for n1 To see why this logic works let's try using these two steps on simple examples. Suppose that you have a million boxes all in a single file straight line and you are standing at the front of that line. Next suppose I state the following two facts about the boxes: Fact 1: The box at the front of the line is empty. Fact 2: If a given box is empty, then the box directly behind that box must also be empty. Q: What can you conclude from this? A: You know that the first box is empty. Now by applying “Fact 2” it must be the case that the box directly behind the first box (i.e. the second box) is empty. Applying “Fact 2” again means that the box directly behind the second box (i.e. the third box) is empty. If you continue in this way you will see that “Fact 1” and “Fact 2” together imply that all of the boxes are empty. Notice that if it was originally your goal to prove that all boxes are empty, then it would have been sufficient for you to prove “Fact 1” and “Fact 2” are both true. The exact same type of logic works in mathematical induction, except instead of dealing with boxes you are dealing with natural numbers, and instead of proving that all boxes are empty you are trying to prove a formula works for all natural numbers. You first demonstrate that the formula works for the first number ( n=1 ). Then you show that if the formula works for a given number, n, then it must work for the number directly behind it (i.e. n1 ). Together these two facts imply that the formula must work for all natural numbers. Let's consider an example (on the next page) where we use mathematical induction to prove the Gaussian Sum Formula. Math 1210-007 Mathematical Proofs Fall 2010 Gaussian Sum Formula: The sum of the first n natural numbers can be more easily calculated as follows: n ∑k =1 k =123n= nn1 = P n 2 Proof: Step 1: First we want to show that the formula works when n=1 . The sum of the first integer is 1 ∑k =1 k =1 Using the formula with n=1 gives P 1= 111 2 = =1 2 2 So the formula works in the case when n=1 Step 2: Assume that the formula works for some number n, then show that it must be the case the the formula works for n1 n n1 =P n for some number n. Now lets see what 2 we get when we sum the first n1 integers. So we are assuming that 123n= n ∑k =1 k =123n n1=1n n1= n n1 n1n11 n1= 2 2 The expression on the far right is the result of several steps of algebraic simplification done to the second to last expression. More importantly, notice that the expression on the far right is equivelant to P n1 . Therefore, by assuming that the formula works for a number n, we were able to use this fact to show that the formula must also work for n1 . In other words, we have found that n1 ∑k =1 k =P n1 . Let us step back for a second and summarize what we have proven. We have proven that P 1 is equivalent to the sum of the first integer. We have also proven that if P n is the sum of the first n integers, then P n1 is the sum of the first n1 integers. Therefore, having proven these two essential facts, we can apply the logic of mathematical induction to conclude that the Gaussian sum formula must work for all natural numbers. Q.E.D.