a fost răspuns

Calculati radx daca x = 1/1×2 + 1/2×3 +1/3×4 + .... + 1/80×81​

Răspuns :

Anastasiaciobanu1ask

x=1/1*2+1/2*3+1/3*4+...+1/80*81

x=1/1-1/2+1/2-1/3+1/3-1/4+...+1/80-1/81

Se simplifica si ramanem cu:

x=1/1-1/81 ; amplificam pe 1/1 cu 81

x=81/81-1/81=80/81

radical din x = radical din 80/81 = 4 radical din 5 /9

Triunghiul1ask

Răspuns:

[tex]\sqrt{x} =\frac{4\sqrt{5} }{9}[/tex]

Explicație pas cu pas:

[tex]x=\frac{1}{1\cdot2} +\frac{1}{2\cdot3} +\frac{1}{3\cdot4} +...+\frac{1}{80\cdot81}[/tex]

[tex]x=\frac{1}{k(k+1)} +\frac{1}{k(k+1)} +\frac{1}{k(k+1)} +...+\frac{1}{k(k+1)}[/tex]

[tex]Orice\,\:numar\,\:sub\,\:forma\,\:\frac{1}{k(k+1)}\, \: poate\,\:fi\\scris\,\:ca\,\: \frac{1}{k} -\frac{1}{k+1}[/tex]

[tex]x=\frac{1}{1}-\not{\frac{1}{2} }+\not{\frac{1}{2}} -\not{\frac{1}{3}} +\not{\frac{1}{3} }-\not{\frac{1}{4}} +...+\not{\frac{1}{80}} -\frac{1}{81}[/tex]

[tex]Observam\,\: ca\,\:a\,\:doua\,\:fractie\,\:se\,\:simplifica\,\:cu\\a\,\:treia,\,\:a \,\:patra\,\:cu\,\:a\,\:cincea\,\:si\,\:tot\,\:asa.[/tex]

[tex]\implies x=\frac{1}{1} -\frac{1}{81} \,|\cdot81 \implies x=\frac{81}{81} -\frac{1}{81} =\frac{80}{81}[/tex]

[tex]\implies \sqrt{x} =\sqrt{\frac{80}{81} } =\frac{4\sqrt{5} }{9}[/tex]