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Q1. (Specimen Paper 1H) Prove algebraically that the difference between any two different odd numbers is an even number. ae (Qn41) - (2m!) y 2ntt - 2m 2n-2m =2(n-m) .. even (Total for question = 3 marks) Q2. (March 2013 2H) ‘mana A Ls eich icici | (ii) Show that when x is a whole number 7(2x + 1) + 6(x + 3) is always a multiple of 5 7(2x+1) + 6(x+3) 1Yx+7 + 6x+18 20x+25 va S(Yx+S) «. multiple of S re (Total for Question is 4 marks) Q3. (June 2017 1H) nis an integer greater than 1 Prove algebraically that n2 - 2 - (n - 2)? is always an even number. n?- 2- (n-2)(n-2) sn? 2-(n?-4nt) ent-2- x +4n-Y =Un-6 =2(2n-3) .- even (Total for question = 4 marks) Q4. (June 2012 2H) Prove that (2n + 3)? — (2n- 3) is a multiple of 8 for all positive integer values of n. (2n+3)(2n+3)- (2n-3)(2n -3) \ =Ynr+)2nF49 - (Yn? -12n +9) Unt +12naF- Un + one 24n ff 3(3n) .. multiple of 8 (Total for Question is 3 marks) Q7. (June 2019 1H) Given 5% that n can be any integer such that n > 1, prove that n?— nis never an odd number n-2, col ny, ‘ey n+2. As n and (n#)) are consecutive number then or leaet one of them if even, 7.2 a multiple of 2. Anything multiplied by 2 wil always be we even, *.n*=n 12 never an odd numbet. ( (Total for question = 2 marks) Q8. (June 2011 2H) Prove that (n-1)? +n? + (n+ 1)? =3n?+2 (n-1)(n=1) +12 + (ne (nt!) 2 n?-IQntl+n24+ n> +204) ide Va 3n24+2 V (Total for Question is 2 marks) Q9. (Specimen Paper 3H) H lere are the first five terms of an arithmetic sequence Ly 4 Az én + ] if 13 19 25 31 Prove that the difference between the squares of any two terms of the sequence Is always a 2 n,m both even (énel)?- (6m +!) of 12(n-mn) C3(n4 I-13 multiple of 24 (bn¥1)(6n4l) - (6m+42) (b+!) Aha F-£=Ben v ay = eatibipe of 24 n,m ‘eth odd = 3bn2 +1 2nzt - Bim? = 120s ys alll AN mu It le YAS = 3bn2- 34m* +12n-12m VA mtn ee! V- Fa 4 =(36n +1(2n+1)- (36m? +12m4) = 36(n2- m?) + 12(n-m) n,m different 36(n+m)(n=m) + 12(n=m) Bene € laser) 17 odd (nance rans fa 12ln-m) [3(n4m) 41] (7 o Saligle of. 0z pow Q10. (Noyember 20193H) Bas matt iple of 12 moulBip le of 2/even Bs eos 12 Prove algebraically that the sum of the squares of any two consecutive even numbers is always a multiple of 4 (2n)* + (2n+ 2)" = Une aX (2n+2)(2n+2) =Y4n?+ Yn? + Bnt+4 =Sn24+3nt4 if =4(2n2+2n+!) re multiple of Y vo (Total for question = 3 marks) Q13. (Specimen Paper 2H) Prove algebraically that th i difference betwe ecutive integers is always an odd number en the squares of any two cons g (n#1)? = (n)? =(n41)(n4) = n* =¥*4+2nt] - y= a =2n+] . odd (Total for question = 3 marks) Q14. (Specimen Paper 3H) The product of two consecutive positive integers is added to the larger of the two integers. Prove that the result is always a square number. (niinet) t (nt, n7+2n +! Cnt) (n+) fy squere number / (Total for question = 3 marks) \V ees Ds Q15. (November 2017 4H) nis an integer S an intege: Prove algebre number (n+intl) ino)” Ses Squere num ber (Total for question = 2 marks) Q18. (Specimen Paper 3H) S ; , b, C are positive integers such that a > b > c Nis the largest thi num has the digi an ree digit numb: d ig er that has the digits a Kis the smallest three digit number that has the dite e.6 and c (a) Use algebra t ig lo show that the difference between N and Kis always a multiple of 99 =ax100+ bx10d +C =1000+10b +c g@cxlOCt+tbxlO+oa , \ =100c+ 0b +2 fv- IK = (1000 +10b +¢)- (100+ 106+) 1000 4206 +c - 100c -J8b -2 = 992.- 19% 99%(a-c) .. multiple of 99 (b) Ifa> band b=c will the difference between Nand K still be a multiple of 99? Justify your answer. bis cancel..out,..:..nee.no.effeck.. v.. . (Total for question = 4 marks) 
