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![If [X] denotes the greatest integer less than or equal to x, what is](/forum/styles/gmatclub_light/imageset/icon_post_target_unread.svg "If [X] denotes the greatest integer less than or equal to x, what is"){kind=icon}
25 May 2020, 09:56
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B
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
95%
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Question Stats:
58% (03:22) correct
42%
(02:38)
wrong
based on 507
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If [X] denotes the greatest integer less than or equal to x, what is the value of \([√1] + [√2] + [√3] + ... + [√100]\) ?
A. 615
B. 625
C. 505
D. 5050
E. 5125
Are You Up For the Challenge: 700 Level Questions
A. 615
B. 625
C. 505
D. 5050
E. 5125
Are You Up For the Challenge: 700 Level Questions
27 May 2020, 04:52
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perfect squares from 1 to 100 = 1, 4, 9, 16, 25, 36, 49, 64, 81, 100
the root of any number in between will be a ratio = 1.41, 1.73, 2.24, 2.65......
numbers between 1 and 4 = 2 [exclusive of the two numbers]
numbers between 4 and 9 = 4
numbers between 9 and 16 = 6 and so on
so we can write
[√1]+[√2]+[√3]+...+[√100]
= 1 +1x2 + 2 + 2x4 + 3 +3x6 + 4 + 4x8 + 5 + 5x10 + 6 +6x12 + 7 + 7x14 + 8 + 8x16 + 9 +9x18 + 10
or 1x3 + 2x5 + 3x7 +.........9X19 +10
First 9 terms form a series as where any term is:
Tn = n(2n+1)
= 2n^2 + n
Sum2(Tn) = 2(1^2 + 2^2 + 3^2 + ......+9^2) + (1+2+3+....+9)
= 2(9x10x19)/6 + 45
=570+45 = 615
This is the sum of first 9 terms.
We also have 10th term = 10
So, sum of all terms = 625
Ans B
the root of any number in between will be a ratio = 1.41, 1.73, 2.24, 2.65......
numbers between 1 and 4 = 2 [exclusive of the two numbers]
numbers between 4 and 9 = 4
numbers between 9 and 16 = 6 and so on
so we can write
[√1]+[√2]+[√3]+...+[√100]
= 1 +1x2 + 2 + 2x4 + 3 +3x6 + 4 + 4x8 + 5 + 5x10 + 6 +6x12 + 7 + 7x14 + 8 + 8x16 + 9 +9x18 + 10
or 1x3 + 2x5 + 3x7 +.........9X19 +10
First 9 terms form a series as where any term is:
Tn = n(2n+1)
= 2n^2 + n
Sum2(Tn) = 2(1^2 + 2^2 + 3^2 + ......+9^2) + (1+2+3+....+9)
= 2(9x10x19)/6 + 45
=570+45 = 615
This is the sum of first 9 terms.
We also have 10th term = 10
So, sum of all terms = 625
Ans B
General Discussion
26 May 2020, 00:15
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See the attachment. Kindly let me know if there is any shortcut for this. It took me 4 minutes to solve.
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