greatest Integer equations.

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\hspace{-16}$If $\mathbf{[\sqrt{1}]+[\sqrt{2}]+[\sqrt{3}]+.......+[\sqrt{x^2-1}]=y}$\\\\ Then $\mathbf{(x,y)=}$

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Ketan Chandak ·2012-03-14 18:44:20

in this question we have to observe that difference of consecutive perfect squares are in AP bcoz (x+1)² -x²=2x+1
therefore in y 1 will occur thrice.....2 five times ,3 7 times and so on...and the last term will be added 1 less...

therefore y is the sum of the series whose nth term is (n-1)(2x-1)

therefore using summation formulas....we get...

y=x(x+1)(4x-7) 6+x

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Basic Integral Calculus Operators Symbols Matrix Cases

√ √ || {} [] () → ~ ^ x x x → → ln log lg lim lim cos sin tan cot arcsin arccos arctan

∫ ∫ ∫ ∫ ∫ ∬ ∬ ∬ ∬ ∬ ∭ ∭ ∭ ∭ ∭ ⨌ ⨌ ⨌ ⨌ ⨌

∑ ∑ ∑ ∑ ∑ ∮ ∮ ∮ ∮ ∮ ∏ ∏ ∏ ∏ ∏ ∐ ∐ ∐ ∐ ∐

+ - × ÷ ± ∓ = ∪ ∩ ≠ ∼ ≈ ∅ ∀ ∃ ∈ ∋ ∝ ≤ ≥ ≃ ≅ ≡ < > ≪ ≫ ⊂ ∉ ⊃ ⊆ ⊇ ⇒ ⇐ ⇔ ⇒ ⇐ ⇔ ⇑ ⇓ ⇕ → ← ↔ → ← ↔ ↑ ↓ ↕ ↖ ↗ ↘ ↙ ↪ ↩ ⇀ ↼ ⇁ ↽

° ∂ ∞ α β γ δ θ λ μ ν ξ π ρ σ φ ω ε ƒ κ τ υ Γ Δ Λ Σ Π Ω

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greatest Integer equations.

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