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# If [X] denotes the greatest integer less than or equal to x, what is

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Math Expert
Joined: 02 Sep 2009
Posts: 65062
If [X] denotes the greatest integer less than or equal to x, what is  [#permalink]

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25 May 2020, 08:56
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Question Stats:

45% (03:13) correct 55% (02:20) wrong based on 71 sessions

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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

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Re: If [X] denotes the greatest integer less than or equal to x, what is  [#permalink]

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25 May 2020, 23:15
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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1.PNG [ 46.28 KiB | Viewed 739 times ]

Intern
Joined: 05 Jan 2018
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Re: If [X] denotes the greatest integer less than or equal to x, what is  [#permalink]

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27 May 2020, 03:52
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
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Posts: 3
Re: If [X] denotes the greatest integer less than or equal to x, what is  [#permalink]

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28 May 2020, 08:41
[√1]+[√2]+[√3]+...+[√100]
1- 3 times (for √1, √2, √3)
2 - 5 times ( for √4,√5,√6,√7,√8)
.
.
.
10 - 1 time (for √100)

so sum : { 1*3 + 2*5 + 3*7 +... 9 terms } + 1* 10
= > Sum{n(3+(n-1)*2) + 10 = Sum{n(2n + 1)}|9 terms + 10 => sum(2n^2 + n)|9 terms + 10
=> 570 + 45 + 10 = 625

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B
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Re: If [X] denotes the greatest integer less than or equal to x, what is  [#permalink]

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30 May 2020, 13:37
Bunuel wrote:
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

Let’s determine the number of terms whose values are 1, 2, 3, etc.

Value of 1: [√1], [√2], [√3] → 3 terms

Value of 2: [√4], [√5], …, [√8] → 5 terms

Value of 3: [√9], [√10], …, [√15] → 7 terms

Without continuing the list, we can see that there are 9 terms with the value of 4, 11 with 5, 13 with 6, 15 with 7, 17 with 8 and 19 with 9. Finally, there is 1 more term, the last term, with the value of 10.

Therefore, the sum of all 100 terms is:

1 x 3 + 2 x 5 + 3 x 7 + 4 x 9 + 5 x 11 + 6 x 13 + 7 x 15 + 8 x 17 + 9 x 19 + 10 x 1

3 + 10 + 21 + 36 + 55 + 78 + 105 + 136 + 171 + 10 = 625

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Re: If [X] denotes the greatest integer less than or equal to x, what is   [#permalink] 30 May 2020, 13:37