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f(x) denotes the number of positive integers less than or equal

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by Max@Math Revolution » Wed Feb 19, 2020 1:11 am
[GMAT math practice question]

f(x) denotes the number of positive integers less than or equal to \(\sqrt{x}\) . What is f(1) + f(2) + f(3) +…+ f(50)?

A. 131
B. 217
C. 228
D. 307
E. 339
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Re: f(x) denotes the number of positive integers less than or equal

by Max@Math Revolution » Fri Feb 21, 2020 12:41 am
=>

Since 1 < √2 < √3 < 2, we have f(1) = f(2) = f(3) = 1.
Since 2 < √5 < √6 < √7 < √8 < 3, we have f(4) = f(5) = … = f(8) = 2.
Since 3 < √10 < √11 < √12 < … < √15 < 4, we have f(9) = f(10) = … = f(15) = 3.
Since 4 < √17 < √18 < √19 < … < √24 < 5, we have f(16) = f(17) = … = f(24) = 4.
Since 5 < √26 < √27 < √28 < … < √35 < 6, we have f(25) = f(26) = … = f(35) = 5.
Since 6 < √37 < √38 < √39 < … < √48 < 7, we have f(36) = f(37) = … = f(48) = 6.
Since 7 < √50, we have f(49) = f(50) = 7.

Then, f(1) + f(2) + f(3) + … + f(50) = 1*3 + 2*5 + 3*7 + 4*9 + 5*11 + 6*13 + 7*2 = 3 + 10 + 21 + 36 + 55 + 78 + 14 = 217.

Therefore, B is the answer.
Answer: B
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