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

Question

If   denotes the greatest integer less than or equal to x, what is the value of    +   +   + ... +    ?

A. 615
B. 625
C. 505
D. 5050
E. 5125

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skhemani · Accepted Answer

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  
numbers between 4 and 9 = 4
numbers between 9 and 16 = 6 and so on
so we can write
 + + +...+ 
= 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

sambitspm · Answer

See the attachment. Kindly let me know if there is any shortcut for this. It took me 4 minutes to solve.

Hupadhayay · Answer

+ + +...+ 
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

Ans should be
B

ScottTargetTestPrep · Answer

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

Value of 1:  ,  ,   → 3 terms

Value of 2:  ,  , …,   → 5 terms

Value of 3:  ,  , …,   → 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

Answer: B

mentallyslow · Answer

What is the basic knowledge needed to solve this one? I am not aware why we would be grouping the numbers. Thanks!

Krunaal · Answer

Because   denotes the greatest integer less than or equal to x, we group the numbers that come before any perfect square, for example  \sqrt{4} = 2 , every square root before  \sqrt{4}  will give 1.something, and since it is inside  , as per definition of   it will become integer 1. Similarly, anything from  \sqrt{5}  to  \sqrt{8}  will become 2.

ohhno · Answer

We need to find:
 + + +⋯+ 
where   means the  greatest integer ≤ x  (floor function).
Group the terms
For n2≤x<(n+1)2,
 =n
So:
 
 1 through 3 → floor = 1 →  3 terms   
 4 through 8 → floor = 2 →  5 terms   
 9 through 15 → floor = 3 →  7 terms   
 ...  
 81 through 99 → floor = 9 →  19 terms   
 100=10 →  1 term    
Thus:
3(1)+5(2)+7(3)+⋯+19(9)+10
The general term is:
(2n+1)n
for n=1 to 9.
n=1∑9(2n2+n)+10
We know:
n=1∑9n2=285
and
n=1∑9n=45
Therefore:
2(285)+45+10 =570+45+10 =625
✅  Answer: B. 625

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

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GMAT Problem Solving (PS) Questions

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

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