Consecutive Numbers Investigation
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.
8, 19, 20
92 = 19*19 = 361
8*20 = 360
Difference 1
97, 98, 99
982 = 98*98 = 9604
97*99 = 9603
Difference 1
17, 118, 119
182 = 118*118 = 13924
17*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.
.2, 2.2, 3.2
2.22 = 2.2*2.2 = 4.84
.2*3.2 = 3.84
Difference 1
0.9, 11.9, 12.9
1.92 = 11.9*11.9 = 141.61
0.9*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. I will now hope to show that it works with algebra.
X, X+1, X+2
st *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
7, 19, 21
7*21 = 357
92 = 19*19 = 361
Difference 4
It would appear that it would work every time. I have tried it three times and it is working all right so far.
I will now try decimals
9.9, 11.9, 13.9
9.9*13.9 = 137.61
1.92 = 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
3.6, 15.6, 17.6
3.6*17.6 = 239.36
5.62 = 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
2 = 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 = ...
This is a preview of the whole essay
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
22 = 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
2.92 = 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
32 = 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
1.12 = 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
02 = 10*10 = 100
Difference 25
2, 7, 12
2*12 = 24
72 = 7*7 = 49
Difference 25
9, 14, 19
9*19 = 171
42 = 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
2.12 = 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
2
4
3
9
4
6
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
0, 11
02 = 10*10 = 100
12 = 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
5.9, 17.9
5.92 = 15.9*15.9 = 252.81
7.92 = 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
12 = 11*11 = 121
Difference 57
8, 21
82 = 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
4, 18
42 = 14*14 = 196
82 = 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
4.7, 18.7
4.72 = 14.7*14.7 = 216.09
8.72 = 18.7*18.7 = 349.69
Difference 113.6
6.1, 20.1
6.12 = 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
02 = 10*10 = 100
Difference 75
5, 20
52 = 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
0.72 = 10.7*10.7 = 114.49
Difference 82
5.1, 20.1
5.12 = 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
2X+1
2
4X+4
3
6X+9
4
8X+16
5
0X+25
