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QUESTION 2 When using proof by mathematical induction to show that n(2n—1)(2n +1) is divisible by 3V ne Z™ the inductive step requires proving () (k+1) (B) (k+1) (k+1) (k+1) k +1)(2k)(2k +2) is divisible by 3. ( (2k)(2k +3) is divisible by 3. k +1)(2k +1)(2k +2) s divisible by 3. ( (D) (k+1)(2k +1)(2k+3) is divisible by 3.
QUESTION 16 (6 marks) Given cos(6) =0V n€ Z use mathematical induction to prove 1— (— l)n cos(2n9> cos(@) — cos(30) + cos(59) — ..+ (_ 1)n+1 cos<<2n — 1)0) N 2005(«9)
QUESTION 8 Let P(n) be the proposition that Y (r+1)3 =nx3 Vnez* Which option represents a correct formulation of the assumption that P(k) is true Vk € Z* in a proof using mathematical induction? @A) Y (k+1)3* T = kx3k (B) Z (k+1)35 = nx3 © Y (r+1)3 T =kx3 @) YF (r+1)3 =3
QUESTION 16 (6 marks) Use mathematical induction to prove that 22 +3n—11is divisible by3VneZ .
QUESTION 3 Given n € Z™ for which proposition can the initial statement for mathematical induction be proven? (A) x* - is divisible by (x+ )V (x+y)#0 2 H2 22 > | 2 B) 12422432+ 4n :gn(Zn +3n—1) ©) (n+1)* +(n+2) is divisible by 3 ! _ B ® 2 (2r-1)(2r+1) n+l
QUESTION 4 When using proof by mathematical induction to prove De Moivre’s theorem expressed as (rcis(@))n = r cis(n0)Vn € Z* which statement would be correct in the proof of the inductive step? (A) (reis(8)) = rrcis(k6) i 01s(k+ 6) k+1 ( (reis(0)) = ©) (r01s(9))k+1= eis(k6+1) (reis(9)) c1s((k+1)0)
QUESTION 18 (5 marks) It is proposed that the following expression is divisible by (1 + cis(@)) forneZ (1 + cis(@)) = 0. 2n+l > cis(r6) =0 Evaluate the reasonableness of the proposition.
QUESTION 15 (6 marks) Use mathematical induction to prove that 22 — 3n — 1 is divisible by 3 forn € Z*.
QUESTION 5 When using proof by mathematical induction to show that (1 + i)* = (—4) where n € Z* the inductive step requires the proof of A) A+d*=(D (B) (1+D* = (-4 (C) (14 )H+1 = (—4)k+1 (D) (1 + 0)*+4 = (—4)k+1
QUESTION 2 Consider the proof of the following proposition using mathematical induction. 3 rr el) =Sn(n +1 +2) VneZ* r=l An appropriate assumption statement within the proof is k (A) Yo k(k+)= kk +k +2) r=l k (B) > k(k+1) = 5n(n+ (+2) r=l k (C) Dr(r+)= = kk +k +2) r=] k 1 (D) Yir(r+) =3n(n+ n+ 2) r=1
QUESTION 15 (5 marks) The sum of a geometric progression with n terms where the first term is | and the common ratio is 7 is given by nN ltertrPers...gr = + (for r #1). r- Prove that this rule is true Vn € Z* using mathematical induction by completing the steps of the proof as indicated. a) Initial statement: [1 mark] Assuming the rule is true for n = k rk] l4¢r¢r°4¢re4+.i¢r%t = 1 (r 41) b) Inductive step: [3 marks] c) Conclusion: [1 mark]
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