Number Theory
Aren't positive integers the damn'dest things? I just noticed today the sum over (1,2,3,4,5,6,7,8,9,10,9,8,7,6,5,4,3,2,1) was 100. And so I tried for (1,2,3,4,5,4,3,2,1) and alas the sum was 25. So here's my proof by MI for the "Mirrored Square" (I'm sure there's some official term for this numerical phenomenon buried in the number theory literature):
WTS
For all positive n:
n^2 = 1 + ... + (n-1) + n + (n-1) + ... + 1
-------------------------------------------
Is it true for 1 and 2?
1 = 1^2
2^2 = 4 = 1 + 2 + 1
Yes.
Let it be true for some positive k-1.
(k-1)^2 = 1 + ... + (k-2) + (k-1) + (k -2) + ... + 1
Showing it is true for k also:
k^2 = ((k - 1) + 1)^2
= (k - 1)^2 + 2(k - 1) + 1
= (k - 1)^2 + k + (k -1)
= 1 + ... + (k -1) + k + (k - 1) + ... + 1
--
EP
WTS
For all positive n:
n^2 = 1 + ... + (n-1) + n + (n-1) + ... + 1
-------------------------------------------
Is it true for 1 and 2?
1 = 1^2
2^2 = 4 = 1 + 2 + 1
Yes.
Let it be true for some positive k-1.
(k-1)^2 = 1 + ... + (k-2) + (k-1) + (k -2) + ... + 1
Showing it is true for k also:
k^2 = ((k - 1) + 1)^2
= (k - 1)^2 + 2(k - 1) + 1
= (k - 1)^2 + k + (k -1)
= 1 + ... + (k -1) + k + (k - 1) + ... + 1
--
EP