For any natural number n, let aₙ be the largest integer not exceeding √n. Then the value of a₁ + a₂ + ⋯ + a₅₀ is
Question:
For any natural number n, let aₙ be the largest integer not exceeding √n. Then the value of a₁ + a₂ + ⋯ + a₅₀ is
For any natural number n, let aₙ be the largest integer not exceeding √n. Then the value of a₁ + a₂ + ⋯ + a₅₀ is
Options
Answer: 217
Explanation:
aₙ = ⌊√n⌋. For n = 1: a₁ = 1. For n = 2,3: aₙ = 1. For n = 4,5,...,8: aₙ = 2. For n = 9,...,15: aₙ = 3. For n = 16,...,24: aₙ = 4. For n = 25,...,35: aₙ = 5. For n = 36,...,48: aₙ = 6. For n = 49,50: aₙ = 7. Sum = 1(3) + 2(5) + 3(7) + 4(9) + 5(11) + 6(13) + 7(2) = 3 + 10 + 21 + 36 + 55 + 78 + 14 = 217.
Explanation:
aₙ = ⌊√n⌋. For n = 1: a₁ = 1. For n = 2,3: aₙ = 1. For n = 4,5,...,8: aₙ = 2. For n = 9,...,15: aₙ = 3. For n = 16,...,24: aₙ = 4. For n = 25,...,35: aₙ = 5. For n = 36,...,48: aₙ = 6. For n = 49,50: aₙ = 7. Sum = 1(3) + 2(5) + 3(7) + 4(9) + 5(11) + 6(13) + 7(2) = 3 + 10 + 21 + 36 + 55 + 78 + 14 = 217.
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