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Some Triangles Sum 2014 (Posted on 2014-08-12) Difficulty: 3 of 5
Determine all possible sequences of consecutive triangular numbers whose sum is precisely 2014.

Extra Challenge: A non computer program based method.

No Solution Yet Submitted by K Sengupta    
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Solution Possible Solution | Comment 1 of 5

Consider a 'sequence of consecutive triangular numbers' as the difference of two sequences, both starting at 1, and ending at m and n respectively.

Then 1/2n(n+1)-1/2m(m+1) = 1/8 (2n+1)^2-1/8(2m+1)^2 , with 0<m<n, describes all possible such truncated sequences.

So (2n+1)^2-(2m+1)^2 = 8*2014 = 16112

(2n-2m)(2n+2m+2) = 16112

(n-m) (m+n+1) = 4028 = 2^2*19*53

It's then just a matter of plugging in the factors:

n-m=1, n+m+1=4028; m=2013,n=2014   
n-m=4, n+m+1=1007; m=501,   n=505   
n-m=19, n+m+1=848; m=96   n=115   
n-m=53, n+m+1= 76; m=11,   n=64   
There are no other positive integer solutions.

Edited on August 12, 2014, 3:31 pm
  Posted by broll on 2014-08-12 15:26:33

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