\n
Problema [tex]2)[\/tex] Use o resultado anterior para provar que [tex]\\sqrt{2}\\lt 1,41421357.[\/tex]<\/p>\n \n<\/div>\n \n Solu\u00e7\u00e3o<\/p>\n \n \t[tex]1)[\/tex] \t<\/p>\n [tex]\\qquad x=\\dfrac{x^2+1}{198}[\/tex] Fazendo [tex]\\boxed{\\alpha=99 + 70\\sqrt{2}}[\/tex] e [tex]\\boxed{\\beta=99-70\\sqrt{2}}[\/tex], temos que [tex]\\boxed{\\alpha + \\beta=198}\\,[\/tex] e [tex]\\,\\boxed{\\alpha \\cdot \\beta=1}\\,.[\/tex] (Perceba que n\u00e3o h\u00e1 necessidade de encontrarmos as ra\u00edzes da equa\u00e7\u00e3o [tex]x^2-198x+1=0[\/tex] para estabelecermos a soma e o produto dessas ra\u00edzes.)<\/span> [tex]\\qquad \\alpha=\\dfrac{\\alpha^2+1}{198}\\gt \\dfrac{1}{198}\\quad[\/tex] e [tex]\\quad\\beta=\\dfrac{\\beta^2+1}{198}\\gt \\dfrac{1}{198}[\/tex], [tex]\\qquad \\alpha=198-\\beta \\lt 198-\\dfrac{1}{198}=197,99494949\\cdots\\quad [\/tex] e [tex]\\quad \\beta=198-\\alpha \\lt 198-\\dfrac{1}{198}=197,99494949\\cdots\\,[\/tex], Logo, por[tex]\\textcolor{#800000}{(i)}[\/tex] e [tex]\\textcolor{#800000}{(ii)}[\/tex], segue que [tex]\\fcolorbox{black}{#eee0e5}{$\\dfrac{1}{198} \\lt\\alpha\\;,\\; \\beta \\lt 197,99494949\\cdots\\, $}\\,.[\/tex]<\/p>\n [tex]2)[\/tex] [tex] \\quad \\sqrt{2}=\\dfrac{\\alpha-99}{70}\\lt \\dfrac{197,99494949\\cdots -99}{70} \\approx 1,41421356 \\lt 1,41421357.[\/tex] Solu\u00e7\u00e3o elaborada pelos Moderadores do Blog<\/strong>.<\/p>\n<\/div>\n \n","protected":false},"excerpt":{"rendered":" Problema (Indicado a partir do 2\u00ba ano do E. M.) (Retirado de uma apostila de problemas de olimp\u00edadas do Col\u00e9gio Batista Santos Dumont \u2013 Fortaleza) [tex]1)[\/tex] Prove que as ra\u00edzes da equa\u00e7\u00e3o [tex]x=\\dfrac{x^2+1}{198}[\/tex] est\u00e3o entre [tex]\\dfrac{1}{198}[\/tex] e [tex]197,99494949\\cdots\\;.[\/tex] [tex]2)[\/tex] Use o resultado anterior para provar que [tex]\\sqrt{2}\\lt 1,41421357.[\/tex] Solu\u00e7\u00e3o [tex]1)[\/tex] Inicialmente, vamos resolver a equa\u00e7\u00e3o … <\/p>\n
\n (Indicado a partir do 2\u00ba ano do E. M.)<\/span><\/p>\n
\n(Retirado de uma apostila de problemas de olimp\u00edadas do Col\u00e9gio Batista Santos Dumont<\/strong> \u2013 Fortaleza) <\/span>
\n[tex]1)[\/tex] Prove que as ra\u00edzes da equa\u00e7\u00e3o [tex]x=\\dfrac{x^2+1}{198}[\/tex] est\u00e3o entre [tex]\\dfrac{1}{198}[\/tex] e [tex]197,99494949\\cdots\\;.[\/tex]<\/p>\n
\n\n
\n[tex]\\qquad 198x=x^2+1[\/tex]
\n[tex]\\qquad x^2-198x+1=0[\/tex]
\n[tex]\\qquad x=\\dfrac{198 \\pm \\sqrt{(-198)^2-4\\cdot1\\cdot1}}{2}=\\dfrac{198 \\pm \\sqrt {39200}}{2}\\\\
\n\\qquad x=\\dfrac{198 \\pm 140\\sqrt{2}}{2}=99 \\pm 70\\sqrt{2}.[\/tex]<\/p>\n
\nObserve que, como [tex]\\alpha[\/tex] e [tex]\\beta[\/tex] s\u00e3o solu\u00e7\u00f5es da equa\u00e7\u00e3o [tex]x=\\dfrac{x^2+1}{198}[\/tex], segue que:<\/p>\n
\nou seja,
\n[tex]\\qquad \\alpha\\gt \\dfrac{1}{198}\\quad[\/tex] e [tex]\\quad\\beta\\gt \\dfrac{1}{198}.\\qquad \\textcolor{#800000}{(i)}[\/tex]<\/p>\n
\nou seja,
\n[tex]\\qquad \\alpha \\lt 197,99494949\\cdots\\quad [\/tex] e [tex]\\quad \\beta\\lt 197,99494949\\cdots\\,.\\qquad \\textcolor{#800000}{(ii)}[\/tex] <\/ul>\n
\nComo [tex]\\alpha = 99+70\\sqrt{2}[\/tex], segue que:<\/p>\n
\nPortanto, [tex]\\fcolorbox{black}{#eee0e5}{$ \\sqrt{2}\\lt 1,41421357$}\\,.[\/tex]<\/p>\n
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