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

Home > Just Math
Structured Sequential Square Root Sum (Posted on 2010-12-29) Difficulty: 3 of 5
Derive a formula for evaluating the following expression in terms of n, given that n is a positive integer.

Σi = 1 to n2 ([√ i] + <√ i>)

Note: [x] is the greatest integer ≤ x, and <x> is the least integer ≥x.

No Solution Yet Submitted by K Sengupta    
No Rating

Comments: ( Back to comment list | You must be logged in to post comments.)
Hints/Tips a start (spoiler) Comment 1 of 1
Hmm.  Wonder why nobody has tackled this yet?  Must be Christmas break.  Well, here's a start:

[√ i] = n if i = n*n, or n-1 if (n-1)(n-1) < i < n*n
<√ i> = n if (n-1)(n-1) < i <= n*n
so
[√ i] + <√ i> = 2n if i = n*n, or (2n-1) if (n-1)(n-1) < i < n*n

Going from (n-1)(n-1) to n*n involves adding 
 (n-1)(n-1) - n*n = 2n-1 terms,
1 whose value is 2n and (2n-2) whose value is (2n-1).

Simplifying, going from (n-1)(n-1) to n*n involves adding 
  2n + (2n-2)(2n-1) = 4n*n - 4n + 2

So, the sought after expression is the same as
Σi = 1 to n (4n*n - 4n + 2)

I'll stop here.  These summations are fairly well known, but it's too much algebra for me right now (plus I am obviously not good at writing exponents on perplexus. The solution involves 4th powers, and I would embarrass myself writing n*n*n*n where I mean n to the 4th power).




  Posted by Steve Herman on 2010-12-30 12:36:09
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 (3)
Unsolved Problems
Top Rated Problems
This month's top
Most Commented On

Chatterbox:
Copyright © 2002 - 2017 by Animus Pactum Consulting. All rights reserved. Privacy Information