1
2 3
4 5 6
7 8 9 10
11 12 13 14 15
16 17 18 19 20 21
22 23 24 25 26 27 28
etc...
Show that adding all the numbers that are not crossed off from 1 to the end of a line, the result is a 4^{th} power.
Example: 1+4+5+6=16=2^{4}
1, 1+3*5,1+3*5+5*13,1+3*5+5*13+7*25,...
First part of each term added is (2n1), second part is (2n(n1)+1), so the whole thing is: sum 1 to n (2n1)(2n(n1)+1) = n^4.
I'm sure there are neater ways of doing it.
Edited on June 28, 2016, 12:44 am

Posted by broll
on 20160628 00:39:20 