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[준]

m이 정수일때 루트a - 2< m < 루트a + 1 을 만족하는 정수a의개수는 74개 이다 m의 값을 구하는 과정을 서술하시오

[notscience]

m=12?

[notscience]

We are given
sqrt(a) - 2 < m, (1)
and
m < sqrt(a) + 1. (2).

From (1), we have m+2 > 0, and thus m-1 > - 3, that is, m-1 >= -2.

We will consider Cases (I) m-1>=0, and (II) 0 > m-1 >= -2 separately.

[notscience]

Bare in mind that a is nonnegative, since a is an integer inside square root.

[notscience]

(I) m-1 >= 0

Note that (1) is equivalent to a < (m+2)^2.
Also note that (2) is equivalent to (m-1)^2 < a, since m-1 >=0.

Thus the condition [(1) and (2)] is equivalent to

(m-1)^2 < a < (m+2)^2.

Since these inequalities hold for 74 integers,

76 > (m+2)^2 - (m-1)^2 >= 74.

Hence

76 > 6m + 3 >= 74.

Therefore m=12.

[notscience]

(II) 0 > m-1 >= -2

In this case, we have 3 > m+2 >= 1. Hence from (1), we have sqrt(a) < 3. So a < 9. This inequality does not hold for 74 nonnegative integers.

[notscience]

By the arguments in Cases (I) and (II), m=12.

[준]

^^감사합니다~~~