Consecutive Numbers Investigation Essay Example
Consecutive Numbers Investigation Essay Example

Consecutive Numbers Investigation Essay Example

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  • Pages: 5 (1215 words)
  • Published: September 15, 2017
  • Type: Case Study
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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 wou

...

ld 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. I will now hope to show that it works with algebra.

X, X+1, X+2

1st *3rd = X*(x+2) = X2=2X

2nd squared = (X+1)2 = (X+1)(X+1)

(X +1)(X+1) = X2 + 1 + 1X + 1X

= X2 + 2X + 1

The only difference is +1. It shows that the difference will always be 1.

I am now going to see what happens if I make the gap 2.

Gap 2

3, 5, 7

3*7 = 21

52 = 5x5 = 25

Difference 4

5, 7, 9

5*9 = 45

72 = 7*7 = 49

Difference 4

17, 19, 21

17*21 = 357

192 = 19*19 = 361

Difference 4

It would appear that it would work every time. I have tried it three times and it is workin

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all right so far.

I will now try decimals

9.9, 11.9, 13.9

9.9*13.9 = 137.61

11.92 = 11.9*11.9 = 141.61

Difference 4

7.1, 9.1, 11.1

7.1*11.1 = 78.81

9.12 = 9.1*9.1 = 82.81

Difference 4

13.6, 15.6, 17.6

13.6*17.6 = 239.36

15.62 = 15.6*15.6 = 243.36

Difference 4

It would appear that it works using decimals.

I will now try negative numbers.

-1, 1, 3

-1*3 = -3

12 = 1*1 = 1

Difference 4

-10, -8, -6

-10*-6 = 60

-82 = -8*-8 = 64

Difference 4

I have found out that it also works with negative numbers.

I will now hope to show that it works with algebra.

X, X+2, X+4

X*(X+4)= X2 + 4X

(X+2)2 = (X+2)(X+2)=X2 + 4 + 2x + 2 X

= X2 + 4X + 4

The only difference is +4. It shows that the difference will always be 4.

Gap 3

5, 8, 11

5*8 = 55

82 = 8*8 = 64

Difference 9

2, 5, 8

2*8 = 16

52 = 5*5 = 25

Difference 9

9, 12, 15

9*15 = 135

122 = 12*12 = 144

Difference 9

It appears that it will work every time. I have tried it three times and it works all right so far.

I will now try decimals.

2.5, 5.5, 8.5

2.5*8.5 = 21.25

5.52 = 5.5*5.5 = 30.25

Difference 9

4.2, 7.2, 10.2

4.2*10.2 = 42.84

7.22 = 7.2*7.2 = 51.84

Difference 9

9.9, 12.9, 15.9

9.9*15.9 = 157.41

12.92 = 12.9*12.9 = 166.41

Difference 9

It would appear that it works using decimals.

I will now try negative numbers.

-10, -7, -4

-10*-4 = 40

-72 = -7*-7 = 49

Difference 9

-1, 2, 5

-1*5 = -5

22 = 2*2 = 4

Difference 9

I have found out that it also works with negative numbers.

I will hope to show that it works with algebra.

X, X+3, X+6

X*(X+6)=X2 + 6X

(X+3)2 =(X=3)(X+3) =X2 + 6+ 3X + 3X

=X2 + 6X + 9

The only difference is +9. This shows that the difference will always be nine.

Gap 4

5, 9,

13

5*13 = 65

92 = 9*9 = 81

Difference 16

2, 6, 10

2*10 = 20

62 = 6*6 = 36

Difference 16

9, 13, 17

9*17 = 153

132 = 13*13 = 169

Difference 16

It appears that it will work every time. I have tried it three times and it works all right so far.

I will now try decimals.

2.5, 6.5, 10.5

2.5*10.5 = 26.25

6.52 = 6.5*6.5 = 42.25

Difference 16

4.2, 8.2, 12.2

4.2*12.2 = 51.24

8.22 = 8.2*8.2 = 67.24

Difference 16

7.1, 11.1, 15.1

7.1*15.1 = 107.21

11.12 = 11.2*11.2 = 123.21

Difference 16

It would appear that it works using decimals.

I will now try negative numbers.

-1, 3, 7

-1*7 = -7

32 = 3*3 = 9

Difference 16

-10, -6, -2

-10*-2 = 20

-62 = 6*6 = 36

Difference 16

I have found out that it also works with negative numbers.

I will now hope to show that it works with algebra.

X, X+4, X+8

X*(X+8) = X2 = 8X

(X+4)2=(X+4)(X+4)=X2+8+4X+4X

=X2+8X+16

The only difference is +16. It shows that the difference will always be 16.

Gap 5

5, 10, 15

5*15 = 75

102 = 10*10 = 100

Difference 25

2, 7, 12

2*12 = 24

72 = 7*7 = 49

Difference 25

9, 14, 19

9*19 = 171

142 = 14*14 = 196

Difference 25

It appears that it will work every time. I have tried it three times and it works all right so far.

I will now try decimals.

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

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.

2.5, 3.5

2.52 = 2.5*2.5 = 6.25

3.52 = 3.5*3.5 = 12.25

Difference 6

4.5, 5.5

4.52 = 4.5*4.5 = 20.25

5.52 = 5.5*5.5 = 30.25

Difference 10

7.2, 8.2

7.22 = 7.2*7.2 = 51.84

8.22 = 8.2*8.2 = 67.24

Difference 15.4

I have found out that it also works with decimals.

I will now try negative numbers.

-5, -6

-52 = -5*-5 = 25

-62 = -6*-6 = 36

Difference 11

-9, -10

-92 = -9*-9 = 81

-102 = -10*-10 = 100

Difference 19

-2, -3

-22 = -2*-2 = 4

-32 = -3*-3 = 9

Difference 5

It would appear that it works using negative numbers.

I will now hope to show that it works with using algebra

X, X+1

X2 (X+1)2

(X+1)2(X+1)

X2 + 2 + 1X+2X

X2+2X+3

Difference 2X+1

I have noticed that the two consecutive numbers are the difference when added together.

Gap 2

2, 4

22 = 2*2 = 4

42 = 4*4 = 16

Difference 12

6, 8

62 = 6*6 = 36

82 = 8*8 = 64

Difference 28

3, 5

32 = 3*3 = 9

52 =5*5 = 25

Difference 16

I will now try decimals.

2.5, 4.5

2.52 = 2.5*2.5 = 6.25

4.52 = 4.5*4.5 = 20.25

Difference 14

5.5, 7.5

5.52 = 5.5*5.5 = 30.25

7.52 = 7.5*7.5 = 56.25

Difference 26

15.9, 17.9

15.92 = 15.9*15.9 = 252.81

17.92 = 17.9*17.9 = 320.41

Difference 67.6

It would appear that it works using decimals.

I will now try negative numbers.

-7, -9

-72 =

-7*-7 = 49

-92 = -9*-9 = 81

Difference 32

-15, -17

-152 = -15*-15 = 225

-172 = -17*-17 = 289

Difference 64

I have found out that it also works with negative numbers.

I will now hope that it works using algebra.

X, X+2

X2 (X+2)2

(X+2)2(X+2)

X2+4+2X+2X

X2+4X+4

Difference 4X+4

The pattern here is, add two consecutive numbers together and then double them to get your difference.

Gap 3

5, 8

52 = 5*5 = 25

82 = 8*8 = 64

Difference 39

8, 11

82 = 8*8 = 64

112 = 11*11 = 121

Difference 57

18, 21

182 = 18*18 = 324

212 = 21*21 = 441

Difference 117

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.

2.4, 5.4

2.42 = 2.4*2.4 = 5.76

5.42 = 5.4*5.4 = 29.16

Difference 23.4

21.6, 24.6

21.62 = 21.6*21.6 = 466.56

24.62 = 24.6*24.6 = 605.16

Difference 138.6

58.9, 61.9

58.92 = 58.9*58.9 = 3469.21

61.92 = 61.9*61.9 = 3831.61

Difference 362.4

I seems to work with decimals I will now try negative numbers.

-99, -102

-992 = -99*-99 = 9801

-1022 = -102*-102 = 10404

Difference 603

I have found out that it also works with negative numbers.

I will now hope to show that it works with algebra.

X, X+3

X2 (X+3)

(X+3)2(X+3)

X2+6+3X+6X

X2+6X+9X

Difference 6X+9

Instead of adding the consecutive numbers together and multiplying by 2 you multiply it by 3.

Gap 4

5, 9

52 = 5*5 = 25

92= 9*9 = 81

Difference 56

14, 18

142 = 14*14 = 196

182 = 18*18 = 324

Difference 128

30, 34

302 = 30*30 = 900

342 = 34*34 = 1156

Difference 256

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 am now going to try decimals.

5.0, 9.0

5.02 = 5.0*5.0 =

25

9.02 = 9.0*9.0 = 81

Difference 56

14.7, 18.7

14.72 = 14.7*14.7 = 216.09

18.72 = 18.7*18.7 = 349.69

Difference 113.6

16.1, 20.1

16.12 = 16.1*16.1 = 259.21

20.12 = 20.1*20.1 = 404.01

Difference 144.8

It would appear that it works with decimals.

I will now try negative numbers.

-6, -10

-62 = -6*-6 = 36

-102 = -10*-10 = 100

Difference 64

-42, -46

-422 = -42*-42 = 1764

-462 = -46*-46 = 2116

Difference 352

-23, -27

-232 = -23*-23 = 529

-272 = -27*-27 = 729

Difference 200

I have found out that it also works with negative numbers.

I will now hope to show that it works with algebra.

X, X+4

X2 (X+4)2

(X+4)2(X+4)

X2+4X+4X+16

X2+8X+16

Difference 8X+16

Instead of adding the consecutive numbers together and multiplying by 3 you multiply it by 4.

Gap 5

5, 10

52 = 5*5 = 25

102 = 10*10 = 100

Difference 75

15, 20

152 = 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

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