Consecutive Numbers Investigation

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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.
Join now!


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 = ...

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