Let us denote the square root of x rounded to the nearest integer by N(x).
Accordingly N(19) is equal to 4, since √(19) is less than 4.5, but N(23) is 5, because √(23) exceeds 4.5.
Using only p&p, and without a calculator, evaluate the value of this expression:
1 1 1 1
---- + ---- + ---- + ........ + -------
N(1) N(2) N(3) N(2550)
*** Please feel free to use computer program/Excel solver aided methodology for purpose of verification.
The denominators grow slowly, but how many of a given size denominator should there be?
How many 1/n are there for a given n? One for each whole number between (n-.5)^2 and (n+.5)^2.
(n+.5)^2 - (n-.5)^2 = 2n.
So for each 1/n there are 2n copies. These will sum to 2.
sqrt(2550)=50.497 and sqrt(2551)=50.507 so 2550 is the last of the 1/50 fractions. (note also 50.5^2=2550.25)
So there are 50 sums of 2. The answer is 100.
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Posted by Jer
on 2023-05-05 11:49:05 |
Solution
