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| Subtract Half, Get GI? (Posted on 2007-03-20) |
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If ∫o[u]x dx= ∫ou[x] dx, u>0, and
[x] denotes the greatest integer < = x, is it necessarily true that u=[u]+ 1/2?
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Submitted by K Sengupta
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Rating: 2.0000 (1 votes)
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Solution:
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(Hide)
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L.H.S.
= Integral (x^2/2) x=0 to [u]
= [u]^2/2......(i)
If [u] = n,
then RHS
= Integral [x] dx; x = 0 to u
= (Sum (r = 0 to n-1)(Integral [x]dx; x = r to r+1)) + (Integral [x] dx; x= n to u )
= (Sum (r = 0 to n-1)(r*Integral dx; x = r to r+1)) + n(u-n)
= (Sum (r = 0 to n-1)(r*(r+1-r)) + n(u-n)
= n(n-1)/2 + nu - u^2
= nu - n/2 - n^2/2 .....(ii)
So, from (i) and (ii):
n^2/2 = nu - n/2 - n^2/2
Or, nu = n^2 + n/2
Or, u = n+ 1/2 = [u] + 1/2, so that:
u - [u] = 1/2, so that:
u = [u] + 1/2
Consequently, u = [u]+1/2 unless u<1 and u != 1/2.
*********************************************
Also refer to the respective solutions posted by Charlie and Bractals in the Comments Section.
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Comments: (
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Subject |
Author |
Date |
 | Solution | Bractals | 2007-03-21 09:58:20 |
 | background | Charlie | 2007-03-20 15:47:49 |
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