If [x] denotes the greatest integer less than or equal to x, what is the value of $[√1] + [√2] + [√3] + ... + [√100]$ ?

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Correct answer marked below

615

625

Correct

505

5050

5125

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Explanation

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Official Explanation

Official

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:

√2 ≈ 1.41

√3 ≈ 1.73

√5 ≈ 2.24

√7 ≈ 2.65

, etc.

Numbers between 1 and 4 = 2 (exclusive of the two perfect squares).

Numbers between 4 and 9 = 4.

Numbers between 9 and 16 = 6, and so on.

We can write:

[√1] + [√2] + [√3] + ... + [√100] = 1 + 1×2 + 2 + 2×4 + 3 + 3×6 + 4 + 4×8 + 5 + 5×10 + 6 + 6×12 + 7 + 7×14 + 8 + 8×16 + 9 + 9×18 + 10

…

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What this question tests

What it tests

Your ability to evaluate and compose functions, including unusual defined operations.

Common trap

Applying function operations in the wrong order or ignoring domain restrictions.

If [x] denotes the greatest integer less than or equal to x, what is the value of [√1]+[√2]+[√3]+...+[√100][√1] + [√2] + [√3] + ... + [√100][√1]+[√2]+[√3]+...+[√100] ?