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. 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 = 5×5 = 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 working 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
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.
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