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Squaring the triangle (Posted on 2013-01-07) Difficulty: 3 of 5

The number 185136 has the interesting property that it is a triangular number that is also the product of 3 consecutive integers.

I square 185136 and each of the x consecutive numbers following it, and total the sum of all these squares, S.

Then I do the same with 185136+x+1 and the y consecutive numbers following it, until I again reach the same sum, S.

Find positive integer values of x,y,S compliant with the above requirements.

See The Solution Submitted by broll    
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re(2): Bit different with added consideration - short step? | Comment 6 of 7 |
(In reply to re: Bit different with added consideration by broll)

Which triangle number is 185136?

I find n=((√8*T) - 1)/2.

Substituting for T gives n=608 which is twice the value required for the x value of 304.

Again I find that the sum of the first n squares is given by:
    n/3 +  n/2 + n/6.

On that logic it seems that I need the sum of the first (n+304) squares minus the sum of the first n [608] squares to arrive at: 10471153462280.

I'm assuming that is correct without going "through the hoops". 

Having shown in a prior comment that y is x-1 one would do similarly for the ensuing the sum of the next x-1 squares arrives at the same result.

I am extrapolating upon previous findings and internet findings but I feel this is right.    

  Posted by brianjn on 2013-01-08 06:48:29

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