Question

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

Answer 1

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

Answer 2

Răspuns:

$\sqrt{x} =\frac{4\sqrt{5} }{9}$

Explicație pas cu pas:

$x=\frac{1}{1\cdot2} +\frac{1}{2\cdot3} +\frac{1}{3\cdot4} +...+\frac{1}{80\cdot81}$

$x=\frac{1}{k(k+1)} +\frac{1}{k(k+1)} +\frac{1}{k(k+1)} +...+\frac{1}{k(k+1)}$

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

$x=\frac{1}{1}-\not{\frac{1}{2} }+\not{\frac{1}{2}} -\not{\frac{1}{3}} +\not{\frac{1}{3} }-\not{\frac{1}{4}} +...+\not{\frac{1}{80}} -\frac{1}{81}$

$Observam\,\: ca\,\:a\,\:doua\,\:fractie\,\:se\,\:simplifica\,\:cu\\a\,\:treia,\,\:a \,\:patra\,\:cu\,\:a\,\:cincea\,\:si\,\:tot\,\:asa.$

$\implies x=\frac{1}{1} -\frac{1}{81} \,|\cdot81 \implies x=\frac{81}{81} -\frac{1}{81} =\frac{80}{81}$

$\implies \sqrt{x} =\sqrt{\frac{80}{81} } =\frac{4\sqrt{5} }{9}$