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ECS20 Homework 5 Exercise 1 Find these values : a) êë2.4úû b) éê2.4ùú c) êë-3.4úû f ) êé-6.99ùú d) éê-3.4ùú e) éê6.99ùú éê 1 ú é 1 ù 1 ù ê 1 é 1 ùú g) ê + ê úú h) êê ú + ê ú + ú êë 4 û ê 4 ú 2 ú ë 4 ê 4 úû Exercise 2 (proof) a) Show that the following statement is true: “If x is a real number such that x2+2=0, then x4 = -5”. b) Constructive proof: “If x and y are real numbers such that x < y, show that there exists a real number z with x <z < y” Exercise 3 ê 1ú ê 2 ú Let x be a real number. Show that êë3xúû = êë xúû + ê x + ú + ê x + ú ë 3û ë 3 û Exercise 4 êêënxúûú Show that for all strictly positive integer n and for all real number x, ê ú = êë xúû ë n û **Extra credit: Let us consider a generalization of exercise 3. Let x be a real number, and N an integer greater or equal to 3. Show that: 1ú ê 2ú N - 1ú ê ê ëNx û = ëx û + ê x + ú + ê x + ú + … + ê x + Nû ë Nû N úû ë ë
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