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: A tower of cubes is made of 35 cubes in five layers. How many cubes are needed to form a similar tower with n layers?

Here is what I have so far:

1 layer=1 cube, 2 layers=4 cubes, 3 layers=10 cubes, 4 layers=20 cubes, 5 layers=35 cubes...10 layers=220 cubes

This equation works for n, but I don't know why!

n (n+1)(n+2)/6

I really wanted to know.

Please help. Thank you.

Know of what are called Difference Tables?

Take that data

1 1

2 4

3 10

4 20

5 35

and to start the table.

To create the next row, take the difference in successive y's over the difference in successive x's. That would be, in order, (4-1)/(2-1) = 3, (10-4)/(3-2) = 6, (20-10)/(4-3) = 10,

and (35-20)/(5-4) = 15.

The table would then look like

1 1

2 4 3

3 10 6

4 20 10

5 35 15.

To compute the next column, take the difference in successive elements,

but this time vary the x's by 2 units. This would give (6-3)/(3-1) = 1.5,

(10-6)/(4-2) = 4/2 = 2, and (15-10)/(5-3) = 5/2 = 2.5.

The table now looks like

1 1

2 4 3

3 10 6 1.5

4 20 10 2.0

5 35 15 2.5.

To compute the next column, take the last column and the x's 3 units apart.

This gives (2.0-1.5)/(4-1) = 0.5/3 = 1/6 and (2.5-2.0)/(5-2) = 0.5/2 = 1/6.

Since both elements in this column are the same, the last row can be dropped, giving

1 1

2 4 3

3 10 6 1.5

4 20 10 2.0 1/6

To make the equation, take (((1/6)(x-3)+1.5)(x-2) + 3)(x-1) + 1.

This works out to ((x/6 - 1/2 + 1.5)(x-2) + 3)(x-1) + 1.

This is ((x/6 + 1)(x-2) + 3)(x-1) + 1.

That goes to ((x²/6 + 2x/3 - 2 + 3)(x-1) + 1.

Combining -2 + 3 gives 1, so we have (x²/6 + 2x/3 + 1)(x-1) + 1.

That gives x³/6 + x²/2 + x/3-1+1.

The -1+1 cancels, and factoring out 1/6 gives (x³ + 3x² + 2x)/6.

That can be seen to factor into x(x+1)(x+2)/6.

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