Perfect squares and odd numbers

Factual Questions
1

1=square
1+3=4
4+5=9
9+7=16
16+9=25
etc.

If there is a link between perfect squares and odd numbers, is there a link between some number group and perfect cubes?

2

The “link” that you’re showing here (each subsequent square is the previous square plus an odd number) is basically the mathematical expression

(n+1)[sup]2[/sup] = n[sup]2[/sup]+2n+1

(eg “4” + “5” = “9” … the “4” is “2[sup]2[/sup]” and the “5” is “2*2 + 1” )

2n+1 is, obviously, an odd number always. In fact, it’s basically the definition of “odd number”

The cube equivalent is

(n+1)[sup]3[/sup] = n[sup]3[/sup] + 3n[sup]2[/sup]+3n+1.

So if you did the same thing with cubes (and 4-cubes, and 5-cubes) your equivalent of the “odd number” term would be successive values of 3n[sup]2[/sup]+3n+1.

There isn’t really a snappy term like “odd number” for those numbers though…

3

Sum[sub]k=1[/sub][sup]n[/sup] 2k-1 =
2*(Sum[sub]k=1[/sub][sup]n[/sup] k) - n =
2n(n+1)/2 - n =
n[sup]2[/sup]

That is, you’re exactly right: the sum of the first n odd numbers is n[sup]2[/sup]

There’s a formula for the sum of the first n square numbers too…

Sum[sub]k=1[/sub][sup]n[/sup] k[sup]2[/sup] = n*(n+1)*(2n+1)/6 = n[sup]3[/sup]/3 + n[sup]2[/sup]/2 + n/6

Now, we can combine this with the ealier formulas to get

n[sup]3[/sup] = Sum[sub]k=1[/sub][sup]n[/sup] (3k[sup]2[/sup] - 3/2(2k - 1) - 1/6)

4

What is going on here with squares is quite easy to visualize on graph paper:
Start with a square of size 0 and add enough blocks to make it the next square:
*

This gives a square of size 1.

Now add squares to the right and top edges, plus the corner:

o *

This gives a square of size 4.

Now add squares to the right and top sides, plus the corner:

o o *
o o *

This gives a square of size 9.

o o o *
o o o *
o o o *

A square of size 16.

Continue forever…

You can see that the general rule is that to go to the next square, you add the width + the height + 1. This turns out to be 2x + 1. (this is why x^2 + 2x + 1 factored nicely to (x+1)^2 back in algebra)

It is clear from the pictures that the series starts at 1 and always increases by two units; hence the series of odd numbers.

Now for cubes:

To get the next cube, you add three faces, three edges, and a corner cube:
Interval = 3x^2 + 3x + 1. Hence, x^3 + 3x^2 + 3x + 1 factors unto (x + 1)^3

I can’t say that I can make any sense out of that sequence, though.

5

In general, the difference between successive nth powers is given by sum( [sub]n[/sub]C[sub]k[/sub]x[sup]k[/sup], 0 < k < n - 1), where [sub]n[/sub]C[sub]k[/sub] = n!(k!*(n - k)!). This is easy to show once you have the binomial theorem.