Question

Calculati √1x4x6+2x8x12+3x12x18+...100x400x600/1x2x3+2x4x6+3x6x9+...+100x200x300 radical din tot raspunsul trebuie sa fie 2

Answer 1

[tex]\sqrt{\frac{1\cdot4\cdot6+2\cdot8\cdot 12+3\cdot12\cdot18+...+100\cdot 400\cdot 600}{1\cdot 2\cdot 3+2\cdot 4\cdot 6+3\cdot 6\cdot 9+...+100\cdot 200\cdot 300}}=\\ Rezolvam\ ambele\ sume\ separat:\\ 1\cdot4\cdot6+2\cdot8\cdot 12+3\cdot12\cdot18+...+100\cdot 400\cdot 600=\\ 1\cdot 4\cdot 6+2^3\cdot1\cdot4\cdot6+3^3\cdot4\cdot6+...+100^3\cdot4\cdot6=\\ 4\cdot 6(1+2^3+3^3+....+100^3)=\\ \\ 1\cdot 2\cdot 3+2\cdot 4\cdot 6+3\cdot 6\cdot 9+...+100\cdot 200\cdot 300=\\ [/tex]
[tex]1\cdot2\cdot 3+2^3\cdot 2\cdot 3+3^3\cdot 2\cdot 3+...+100^3\cdot 2\cdot 3=\\ 2\cdot3(1+2^3+3^3+...+100^3)=\\ Revenim: \sqrt{\frac{4\cdot 6(1+2^3+3^3+...+100^3)}{2\cdot 3(1+2^3+3^3+...+100^3)}}=\sqrt{\frac{24}{6}}=\sqrt{4}=\bold{2}\\[/tex]