All about flooble | fun stuff | Get a free chatterbox | Free JavaScript | Avatars    
perplexus dot info

Home > Numbers
Squared average equals square (Posted on 2019-08-09) Difficulty: 3 of 5
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:
Login:
Password:
Remember me:
Sign up! | Forgot password


Search:
Search body:
Forums (0)
Newest Problems
Random Problem
FAQ | About This Site
Site Statistics
New Comments (4)
Unsolved Problems
Top Rated Problems
This month's top
Most Commented On

Chatterbox:
Copyright © 2002 - 2021 by Animus Pactum Consulting. All rights reserved. Privacy Information