Question 12Prove that n + 5n is divisible by 6 for all n€ N.Please go on to the next page...MathematicsMTHS311 - Real AnalysisPage 3 of 4Question 13Conjecture a formula for the sum 1/1.3+1/3.5+..+1/(2n -1)(2n+1), and prove yourconjecture using mathematical induction.Question 14Conjecture a formula for the sum of the first n odd natural numbers 1+3+..+(2n – 1),and prove your conjecture using mathematical induction.Question 15Describe the following sets open, closed, both, or neither.(a) (-1,0) U (0, 1)(b) (0, 1) U (0, 1)(c) [-1,이u(0, 1)(d) Z = {..., -2, -1,0, 1, 2, ...}(e) Q = rational numbers(f) {X : |X – 1| < 2}(g) U (-n, )(h) (0, E]Question 16For the following sets find the interior points and boundary points.(a) (0, 1) U {2}(b) {0,1, 2, 3, 4, 5}(c) {1,4, 4. £ - - - }(d) (0, 0)Question 17(a) Show that the union of finitely many closed sets is closed.(b) Give an example showing that the union of infinitely many closed sets may fail tobe closed.Please go on to the next page...MathematicsMTHS311 - Real AnalysisPage 4 of 4Question 18Suppose that {a,} is a sequence of real numbers such that 0 < r, < 1 for all n e N.Define A = U(0, r,). Show that A is bounded.Question 19Find the accumulation points of the following sets (if any). State whether the conditionsof the Bolzano-Weierstrass theorem hold.(a) (2,4) U (4, 5)(b) {(-1)" : n € N} Question 1If a, b eR, prove the following.(a) If a + b = 0, then b = -a,(b) — (-а) — а,(c) (-1)a = -a,(d) (-1)(-1) = 1,(e) -(a + b) = (-a) + (-b),(f) (-a).(-b) = a.b,(g) 1/(-a) = -(1/a),(h) -(a/b) = (-a)/b if b + 0.Question 2If a, b and c are real numbers, prove that(a) If a 0}. Does S have lower bounds? Does S have upper bounds? Doesinf S exist? Does sup S exist? Prove your statements.Question 9Prove by induction that 1/1.2+1/2.3 +..+1/n(n +1) = n/(n + 1) for all ne N.Question 10Prove that 13 + ...+n° = [}n(n +1)]* for all n e N.Question 11

To prove n3 + 5n is divisible by all natural numbers. “Since you have asked multiple question,…

Answered: Question 12 Prove that n + 5n is…

Question 12 Prove that n + 5n is divisible by 6 for all n E N.

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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