SOLUTION: Please help me on this problem: Suppose "k" is some positive interger. Consider numbers of the form √n) where "n" is a positive interger. In terms of "k," how many numbers o

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Question 761962: Please help me on this problem: Suppose "k" is some positive interger. Consider numbers of the form √n) where "n" is a positive interger. In terms of "k," how many numbers of this form differ from "k" bh less than 1? Explain why your answer works in general. (Hint: for this question it may help to try some small values of "k" and search for a pattern in the results. For example, if k= 2 there are 7 numbers of the desired form that differ from "k" by less than one since √2), √3), √4), √5), √6), √7), and √8) are the only such numbers that differ from 2 by less than 1. Now, try k=3 and so forth. After you find a formula think about why your formula works in general.)
Answer by ramkikk66(644) About Me  (Show Source):
You can put this solution on YOUR website!
For any number k,
all values of n in the range (k-1)^2 and (k+1)^2 will satisfy the condition, where sqrt(n) would differ from k by less than 1.
How many numbers are there between k^2 and (k+1)^2?
%28k%2B1%29%5E2+-+k%5E2+=+2%2Ak+%2B+1
How many numbers are there between k^2 and (k-1)^2?
%28k%29%5E2+-+%28k-1%29%5E2+=+k%5E2+-+%28k%5E2+-+2k+%2B+1%29+=+2%2Ak+-+1
Sum of the two = 4%2Ak
However, this list of 4k numbers also includes k twice. So we have to reduce it by 1.
So, total possible values of n where sqrt(n) differs from k by less than 1,
is highlight%284%2Ak+-+1%29
:)

Check a few samples:
For k = 2, the set of n would be sqrt(2) to sqrt(8) = 7 numbers = 4*2 - 1
For k = 3, the set of n would be sqrt(5) to sqrt(15) = 11 numbers = 4*3 - 1
For k = 5, the set of n would be sqrt(17) to sqrt(35) = 19 numbers = 4*5 - 1