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[x] is the greatest integer less than or equal to x. What is the value

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[x] is the greatest integer less than or equal to x. What is the value  [#permalink]

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New post 02 May 2019, 17:51
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E

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Question Stats:

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[GMAT math practice question]

\([x]\) is the greatest integer less than or equal to \(x\). What is the value of \([√1] + [√2] + [√3] + … + [√49] + [√50]\)?

\(A. 115\)

\(B. 195\)

\(C. 217\)

\(D. 288\)

\(E. 360\)

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Re: [x] is the greatest integer less than or equal to x. What is the value  [#permalink]

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New post 02 May 2019, 19:16
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To solve this question, let us evaluate the ranges of roots till 50.
For x between 1 and 3, [root(x)] will always be 1. For x between 4 and 8, it will be 2. Between 9 and 15, it will be 3. Between 16 and 24, it will be 4. Between 25 and 35, it will be 5. Between 36 and 48, it will be 6. 49 and above, it will be 7.

So the sum required = 3*1 + 5*2 + 7*3 + 9*4 + 11*5 + 13*6 + 2*7 = 3 + 10 + 21 + 36 + 55 + 78 + 14 = 217.

Option (C).
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Re: [x] is the greatest integer less than or equal to x. What is the value  [#permalink]

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New post 06 May 2019, 01:37
=>

\([√1] = 1\)
\([√2] = 1\)
\([√3] = 1\)
If \(1 ≤ n < 4\), then \(1 ≤ √n < 2\)and \([√n] = 1\) for \(n = 1, 2, 3\) (\(3\) terms).
If \(4 ≤ n < 9\), then \(2 ≤ √n < 3\) and \([√n] = 2\) for \(n = 4, 5, …, 8 (5(=8-4+1)\) terms).
If \(9 ≤ n < 16\), then \(3 ≤ √n < 4\) and \([√n] = 3\) for \(n = 9, 10, …, 15 (7\) terms).
If \(16 ≤ n < 25\), then \(4 ≤ √n < 5\) and \([√n] = 4\) for \(n = 16, 17, …, 24 (9\) terms).
If \(25 ≤ n < 36\), then \(5 ≤ √n < 6\) and \([√n] = 5\) for \(n = 25, 26, …, 35 (11\) terms).
If \(36 ≤ n < 49,\) then \(6 ≤ √n < 7\) and \([√n] = 6\) for \(n = 36, 37, …, 48 (13\)terms).
If \(49 ≤ n < 64,\) then \(7 ≤ √n < 8\) and \([√n] = 7\) for \(n = 49\) and \(50 (2\) terms).
Therefore,
\([√1] + [√2] + [√3] + … + [√49] + [√50]\)
\(= ( 1 + 1 + 1 ) + ( 2 + 2 + … + 2 ) + ( 3 + 3 + … + 3 ) + ( 4 + 4 + … + 4 ) + ( 5 + 5 + … + 5 ) + ( 6 + 6 + … + 6 ) + ( 7 + 7 )\)
\(= 1*3 + 2*5 + 3*7 + 4*9 + 5*11 + 6*13 + 7*2\)
\(= 3 + 10 + 21 + 36 + 55 + 78 + 14\)
\(= 217\)

Therefore, the answer is C.
Answer: C
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Re: [x] is the greatest integer less than or equal to x. What is the value   [#permalink] 06 May 2019, 01:37
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