Show p (1) is true. 2 Inductive hypothesis (IH): If k 2N is a generic particular such that k n 0, we assume that P(k) is true. The method of mathematical induction for proving results is very important in the study of Stochastic Processes. Mathematical induction includes the following steps: 1 Inductive Base (IB): We prove P(n 0). The principle of mathematical induction states that if for some property P(n), we have thatP(0) is true and For any natural number n, P(n) → P(n + 1) Then For any natural number n, P(n) is true. Induction Examples Question 6. Mathematical Induction 11. Limit 18. Afterwards, will be available in the filing cabinets in the Gates Open Area near the submissions box. Application of Derivatives 21. Mathematical Induction Part Two. Section 1: Introduction (Summation) 3 1. The following example shows how to use mathematical induction to prove a formula for the sum of the first n integers. Mensuration 16. Announcements Problem Set 1 due Friday, October 4 at the start of class. The statement P0 says that p0 = 1 = cos(0 ) = 1, which is true.The statement P1 says that p1 = cos = cos(1 ), which is true. Definite Intergal 23. Indefinite Intergal 22. Let p0 = 1, p1 = cos (for some xed constant) and pn+1 = 2p1pn pn 1 for n 1.Use an extended Principle of Mathematical Induction to prove that pn = cos(n ) for n 0. Function 17. Binomial Theorem 12. Most often, n 0 will be 0;1, or 2. Final Quiz Solutions to Exercises Solutions to Quizzes The full range of these packages and some instructions, should they be required, can be obtained from our web page Mathematics Support Materials. Straight Line 25. 7 Example 1 – Sum of the First n Integers Use mathematical induction to prove that Solution: To construct a proof by induction, you must first identify the property P(n). Principle of mathematical induction for predicates Let P(x) be a sentence whose domain is the positive integers. Further Examples 4. Solution: Step 1. So p (1) is true. Thus the formula is proved. The Principle of Induction 3. Properties & Solution of Triangle, Height & Distance 15. Solution. Step 2. For any n 0, let Pn be the statement that pn = cos(n ). Base Cases. Trigonometric Ratios, Identities & Equations 13. + n 2 = n (n + 1) (2n + 1) / 6, is true for all positive integers n. Example 3. Continuity 19. Problem Set 1 checkpoints graded, will be returned at end of lecture. 3 Inductive Step (IS): We prove that P(k + 1) is true by making use of the Inductive Hypothesis where necessary. Mathematical induction is therefore a bit like a ﬁrst-step analysis for prov-ing things: prove that wherever we are now, the nextstep will al-ways be OK. Then if we were OK at the very beginning, we will be OK for ever. Differentiation 20. Inverse Trigonometric Function 14. mathematical induction and the structure of the natural numbers was not much of a hindrance to mathematicians of the time, so still less should it stop us from learning to use induction as a proof technique. Differentail Equation 24.

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