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 square root of any number in between will be a decimal (e.g.,

√2 ≈ 1.41

√3 ≈ 1.73

√5 ≈ 2.24

, etc.).

Numbers between consecutive perfect squares:

- Between 1 and 4: 2 numbers (2 and 3)

- Between 4 and 9: 4 numbers (5, 6, 7, 8)

- Between 9 and 16: 6 numbers (10–15)

- And so on.

Thus, 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 handling of square and higher roots, including simplifying radicals.

Common trap

Forgetting the ± when taking a square root to solve an equation, or assuming √(a²) = a.

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] ?