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  • Level: GCSE
  • Subject: Maths
  • Word count: 2408

Consecutive Numbers Investigation

Extracts from this document...

Introduction

James Lucas CONSECUTIVE NUMBERS Problem 1 Gap 1 Take three consecutive numbers; square the middle number, multiply the first by the third number. What do you notice? 2, 3, 4 are consecutive numbers they follow on from each other. The next number is one more than. 2, 3, 4 32 = 3*3 = 9 2*4 = 8 Difference 1 The numbers above are consecutive numbers; the difference between them is one. 18, 19, 20 192 = 19*19 = 361 18*20 = 360 Difference 1 97, 98, 99 982 = 98*98 = 9604 97*99 = 9603 Difference 1 117, 118, 119 1182 = 118*118 = 13924 117*119 = 13923 Difference 1 It appears that it will work every time. I have tried it four times and it works all right so far. I will now try decimals. 1.2, 2.2, 3.2 2.22 = 2.2*2.2 = 4.84 1.2*3.2 = 3.84 Difference 1 10.9, 11.9, 12.9 11.92 = 11.9*11.9 = 141.61 10.9*12.9 = 140.61 Difference 1 It would appear that it works using decimals. I will now try negative numbers. -8, -7, -6 -72 = -7*-7 = 49 -8*-6 = 48 Difference 1 -5, -4, -3 -42 = -4*-4 = +16 -5*-3 = +15 Difference 1 I have found out that it also works with negative numbers. ...read more.

Middle

2.5, 7.5, 12.5 2.5*12.5 = 31.25 7.52 = 7.5*7.5 = 56.25 Difference 25 4.2, 9.2, 14.2 4.2*14.2 = 59.64 9.22 = 9.2*9.2 = 84 64 Difference 25 7.1, 12.1, 17.1 7.1*17.1 = 121.41 12.12 = 12.1*12.1 = 146.41 Difference 25 It would appear that it works using decimals. I will now try negative numbers. -1, 4, 9 -1*9 = -9 42 = 4*4 = 16 Difference 25 -10, -5, 0 -10*0 = 0 -52 = -5*-5 = 25 Difference 25 I have found out that it also works with negative numbers. I will now hope to show that it works with algebra. X, X+5, X+10 X*(X+10)= X2=10X (X+5)2=(X+5)(X+5)= X2 + 10 + 5X + 5X = X2 +10X + 25 The only difference is + 25 This shows that the difference will always be 25. Gap Difference 1 1 2 4 3 9 4 16 5 25 Problem 2 Gap 1 Two consecutive numbers square the first, square the second. What do you notice? 5, 6 52 = 5*5 = 25 62 = 6*6 = 36 Difference 11 7, 8 72 = 7*7 = 49 82 = 8*8 = 64 Difference 15 10, 11 102 = 10*10 = 100 112 = 11*11 = 121 Difference 21 I will now try decimals to see if it works the same. ...read more.

Conclusion

15*15 = 225 202 = 20*20 = 400 Difference 175 40, 45 402 = 40*40 = 1600 452 = 45*45 = 2025 Difference 425 It would appear that it works every time. I have tried it three times and it works all right so far. But this time they is a different pattern. I will now try decimals. 5.7, 10.7 5.72 = 5.7*5.7 = 32.49 10.72 = 10.7*10.7 = 114.49 Difference 82 15.1, 20.1 15.12 = 15.2*15.2 = 228.01 20.12 = 20.1*20.1 = 404.01 Difference 176 42.4, 47.4 42.42 = 42.4*42.4 = 1797.76 47.42 = 47.4*47.4 = 2246.76 Difference 449 It would appear that it works using decimals. I will now try negative numbers. -50, -55 -502 = -50*-50 = 2500 -552 = -55*-55 = 3025 Difference 525 -10, -15 -102 = -10*-10 = 100 -152 = -15*-15 = 225 Difference 125 -22, -27 -222 = -22*-22 = 484 -272 = -27*-27 = 729 Difference 245 I have found out that it also works with negative numbers. I will now hope to show that it works using algebra. X, X+5 X2 (X+5)2 (X+5)2(X+5) X2+5X+5X+25 X2+10X+25 Difference 10X+25 Instead of adding the consecutive numbers together and multiplying by 4 you multiply it by 5. Gap Difference 1 2X+1 2 4X+4 3 6X+9 4 8X+16 5 10X+25 ...read more.

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