Sorry,

the prompt must not be the best. You are right: it's 1414. For 1414*1415/2 = 1000405. Where was the error?

: The sum of all numbers from 1 through n equals Sn = n(n+1)/2.
: You are looking for an n such that Sn < 1000000 but
: Sn+1 > 1000000. This number is 1413. 

The right way actually is this: You are looking for n such that S_n > 1000000 but S_n-1 < 1000000.

Getting sqrt(2000000) is approximately 1414.2... So n=1414 How do you get it? (n-1)*n < 2000000 < n*(n+1).

Taking sqrt(2000000) and rounding it to 1414 is only suggestive and must yet be verified. Because we do not know from the above two inequalities whether 1414^2 is more or less than 1000000.

To convince yourself think of 11 instead of 1000000:

1,2,2,3,3,3,4,4,4,4,5,5,5,5,5.

5 is the first n such that n(n+1)/2 > 11. According to my original answer you were supposed to look for the largest n for which still n(n+1)/2 < 11. This would give n = 4.

With apologies from
East Brunswick, NJ, USA

Copyright © 1996-2009 Alexander Bogomolny

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