 All about flooble | fun stuff | Get a free chatterbox | Free JavaScript | Avatars  perplexus dot info  Squared average equals square (Posted on 2019-08-09) Find n consecutive perfect squares (n > 1)whose average is n2.

 No Solution Yet Submitted by Danish Ahmed Khan No Rating Comments: ( Back to comment list | You must be logged in to post comments.) re: computer solution | Comment 2 of 3 | (In reply to computer solution by Charlie)

n=47, x=-68 is a solution, as negative integers also have perfect squares.

Let sum n=1 to n (x+n)^2 = n^3

Then 1/6n(2n^2+n(6x+3)+6x^2+6x+1) = n^3, so

1/6(2n^2+n(6x+3)+6x^2+6x+1) = n^2, or
1/12 (n^2 - 1) + 1/4 (n + 2 x + 1)^2 = n^2

{{n == 1, x == -2}, {n == 1, x == 0}, {n == 47, x == -69}, {n == 47, x == 21}, {n == 2161, x == -3150}, {n == 2161, x == 988}} etc.

A189173 gives the value of n but does not specify the consecutive squares.

Edited on August 9, 2019, 11:08 pm
 Posted by broll on 2019-08-09 22:59:25 Please log in:

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