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Squares

When squaring numbers some interesting patterns are found.  One is that consecutive squares follow a pattern.

N N2 increase
1 1  
2 4 3
3 9 5
4 16 7
5 25 9
6 36 11
7 49 13
8 64 15
9 81 17
10 100 19
11 121 21
12 144 23
13 169 25
14 196 27
15 225 29

From the table you can see that the difference between squares increases by two as you go along.  Another interesting twist is that 29 is 14 + 15 and 27 is 13 + 14.   Check this out in another place: 15 = 8 + 7 and 9 = 4 + 5.

We can show that by looking at n and n+1.  Then, find n * n and (n+1)(n+1) and subtract.

We get and .  The difference is . Again, looking at some examples verifies it: 29 = 2 * 14 +1 and 19 = 2 * 9 +1.

You can see another curious idea at vishnu.htm