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30 Mar 2023, 07:45
Kudos
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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:
65%
(hard)
Question Stats:
65% (03:14) correct
35%
(02:58)
wrong
based on 248
sessions
History
Date
Time
Result
Not Attempted Yet
If f(x) denotes the number of positive integers less than or equal to \(√x\). What is the value of \(f(1) + f(2) + f(3) +…+ f(50)\)?
A. \(131\)
B. \(217 \)
C. \(228\)
D. \(307\)
E. \(339\)
A. \(131\)
B. \(217 \)
C. \(228\)
D. \(307\)
E. \(339\)
30 Mar 2023, 10:16
Kudos
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Bunuel
Let's start by writing a few terms and understanding the pattern -
Attachment:
Screenshot 2023-03-30 223224.jpg [ 44.11 KiB | Viewed 3876 times ]
Inference: The value of f(x) will changes when \(\sqrt{x}\) is an integer and x is a perfect square.
We can write all the perfect squares between 1 and 50
1
4
9
16
25
36
49
Between 1 and 3, both inclusive there are 3 - 1 + 1 = 3 terms.
The value of f(x) for all of these three terms will be constant (i.e. f(x) = 1)
So f(1) + f(2) + f(3) = 3 * 1 = 3
Here 3 ⇒ number of terms.
Between 4 and 8, both inclusive
- Number of terms = 8 - 4 + 1 = 5
- Value of each f(x) = 2
- Total = 5 * 2 = 10
Between 9 and 15, both inclusive
- Number of terms = 15 - 9 + 1 = 7
- Value of each f(x) = 3
- Total = 7 * 3 = 21
Between 16 and 24, both inclusive
- Number of terms = 24 - 16 + 1 = 9
- Value of each f(x) = 4
- Total = 4 * 9 = 36
Between 25 and 35, both inclusive
- Number of terms = 35 - 25 + 1 = 11
- Value of each f(x) = 5
- Total = 11 * 5 = 55
Between 36 and 48, both inclusive
- Number of terms = 48 - 36 + 1 = 13
- Value of each f(x) = 6
- Total = 13 * 6 = 78
Between 49 and 50, both inclusive
- Number of terms = 2
- Value of each f(x) = 7
- Total = 13 * 6 = 14
Total = 3 + 10 + 21 + 36 + 55 + 78 + 14 = 217
Option B
26 Apr 2023, 08:28
Kudos
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We are given -
f(x) = number of positive integers <= √x
Here we have a max perfect square 49 in this range .
Since √50 will only be little bit larger than √49 , their f(x) will include same number of positive integers.
No. of positive integers <= √49 are 1,2,3,4,5,6,7 .
From here we know that only 7 integers will occur for any f(x) where x is between 1 and 50.
We can calculate the number of times each of these seven integers occur .
Keeping in mind , f(x) = number of positive integers <= √x
The number 7 will occur only for f(49) and f(50) i.e 2 times
The number 6 will occur for f(36) , f(37) , f(38) , ....., f(50) , i.e , (50-36+1) = 15 times
The number 5 will occur for functions from f(25) to f(50) , i.e, (50-25+1) = 26 times
The number 4 will occur for functions from f(16) to f(50) , i.e, (50-16+1) = 35 times
The number 3 will occur for functions from f(9) to f(50) , i.e, (50-9+1) = 42 times
The number 2 will occur for functions from f(4) to f(50) , i.e, (50-4+1) = 47 times
The number 1 will occur for functions from f(1) to f(50) , i.e, 50 times
Total = 217 times
Correct Answer - B
Hope this helps.
f(x) = number of positive integers <= √x
Here we have a max perfect square 49 in this range .
Since √50 will only be little bit larger than √49 , their f(x) will include same number of positive integers.
No. of positive integers <= √49 are 1,2,3,4,5,6,7 .
From here we know that only 7 integers will occur for any f(x) where x is between 1 and 50.
We can calculate the number of times each of these seven integers occur .
Keeping in mind , f(x) = number of positive integers <= √x
The number 7 will occur only for f(49) and f(50) i.e 2 times
The number 6 will occur for f(36) , f(37) , f(38) , ....., f(50) , i.e , (50-36+1) = 15 times
The number 5 will occur for functions from f(25) to f(50) , i.e, (50-25+1) = 26 times
The number 4 will occur for functions from f(16) to f(50) , i.e, (50-16+1) = 35 times
The number 3 will occur for functions from f(9) to f(50) , i.e, (50-9+1) = 42 times
The number 2 will occur for functions from f(4) to f(50) , i.e, (50-4+1) = 47 times
The number 1 will occur for functions from f(1) to f(50) , i.e, 50 times
Total = 217 times
Correct Answer - B
Hope this helps.
