44. Use mathematical induction to show that for any . It is a useful exercise to prove the recursion relation (you don’t need induction). Show that nlines in the plane, no two of which are parallel and no three meeting in a point, divide the plane into n2 +n+2 2 regions. 43. Show that 2n n < 22n−2 for all n ≥ 5. The method of mathematical induction for proving results is very important in the study of Stochastic Processes. 3 Inductive Step (IS): We prove that P(k + 1) is true by making use of the Inductive Hypothesis where necessary. This is not obvious from the deﬁnition. 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. 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. Mathematical induction is a proof technique that can be applied to establish the veracity of mathematical statements. The principle of mathematical induction states that if for some property P(n), we have that P(0) is true and For any natural number n, P(n) → P(n + 1) Then For any natural number n, P(n) is true. 1. 2b. 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. 1b. Prove for every positive integer n,that 33n−2 +23n+1 is divisible by 19. Mathematical induction includes the following steps: 1 Inductive Base (IB): We prove P(n 0). +(2n−1) = n2 where n is a positive integer. Prove, using induction, that all binomial coeﬃcients are integers. 2. 2. [9 marks] Prove by induction that the derivative of is . Find an expression for . Mathematical Induction 2008-14 with MS 1a. Prove by induction that for all n ≥ 1, (a) Show that if u 2−2v =1then (3u+4v)2 −2(2u+3v)2 =1. 2c. INDUCTION EXERCISES 2. Most often, n 0 will be 0;1, or 2. 2a. 2 Inductive hypothesis (IH): If k 2N is a generic particular such that k n 0, we assume that P(k) is true. [8 marks] Let , where . [4 marks] Using the definition of a derivative as , show that the derivative of . Further Examples 4. [3 marks] Consider a function f , defined by . 3. The Principle of Induction 3. 45* Prove the binomial theorem using induction. 2. Principle of mathematical induction for predicates Let P(x) be a sentence whose domain is the positive integers.