高中数列加试题求所有正整数A1,A2...An,使得(99/100)=(A0/A1)+(A1/A2)+...+(An-1/An),其中A0=1,[A(k+1)-1]*A(k-1)>=A(k)*A(k)*[A(k)-1],k=1,2,...n
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![高中数列加试题求所有正整数A1,A2...An,使得(99/100)=(A0/A1)+(A1/A2)+...+(An-1/An),其中A0=1,[A(k+1)-1]*A(k-1)>=A(k)*A(k)*[A(k)-1],k=1,2,...n](/uploads/image/z/1159749-45-9.jpg?t=%E9%AB%98%E4%B8%AD%E6%95%B0%E5%88%97%E5%8A%A0%E8%AF%95%E9%A2%98%E6%B1%82%E6%89%80%E6%9C%89%E6%AD%A3%E6%95%B4%E6%95%B0A1%2CA2...An%2C%E4%BD%BF%E5%BE%97%2899%2F100%29%3D%28A0%2FA1%29%2B%28A1%2FA2%29%2B...%2B%28An-1%2FAn%29%2C%E5%85%B6%E4%B8%ADA0%3D1%2C%5BA%28k%2B1%29-1%5D%2AA%28k-1%29%3E%3DA%28k%29%2AA%28k%29%2A%5BA%28k%29-1%5D%2Ck%3D1%2C2%2C...n)
高中数列加试题求所有正整数A1,A2...An,使得(99/100)=(A0/A1)+(A1/A2)+...+(An-1/An),其中A0=1,[A(k+1)-1]*A(k-1)>=A(k)*A(k)*[A(k)-1],k=1,2,...n
高中数列加试题
求所有正整数A1,A2...An,使得(99/100)=(A0/A1)+(A1/A2)+...+(An-1/An),其中A0=1,[A(k+1)-1]*A(k-1)>=A(k)*A(k)*[A(k)-1],k=1,2,...n
高中数列加试题求所有正整数A1,A2...An,使得(99/100)=(A0/A1)+(A1/A2)+...+(An-1/An),其中A0=1,[A(k+1)-1]*A(k-1)>=A(k)*A(k)*[A(k)-1],k=1,2,...n
[A(k+1)-1]*A(k-1)>=A(k)*A(k)*[A(k)-1] (a2-1)*a0=a1^2(a1-1) a0/a1=a1(a1-1)/(a2-1)=a1a2(a1-1)/a2(a2-1)=(a1/a2)*[a2(a1-1)/(a2-1)] a1/a2=(a0/a1)*(a2-1)/a2(a1-1) a2/a3=(a1/a2)*(a3-1)/a3(a2-1)=(a0/a1)*(a3-1)/a2a3(a1-1) a3/a4=(a2/a3)*(a4-1)/a4(a3-1)=(a0/a1)*(a4-1)/a2a3a4(a1-1) ...a(n-1)/an=[a(n-2)/a(n-1)]*(an-1)/an[a(n-1)-1] =(a0/a1)*(an-1)/a2a3...an(a1-1) 1/a2a3...an=[a(n-1)/an]*(a1/a0)*(a1-1)/(an-1) a0/a1+a1/a2+...+a(n-1)/an =(a0/a1)/(a1-1)[a1-1+(a2-1)/a2+(a3-1)/a2a3+...+(an-1)/a2a3...an] =a0/[a1(a1-1)]*[a1-1+1-1/a2+1/a2-1/a2a3+1/a2a3-1/a2a3a4+...+1/a1a2...a(n-1)-1/a2a3...an] =a0/[a1(a1-1)]*[a1-1/a2a3...an] =[1/(a1-1)][(a1a2...an-1)/(a1a2...an)] =99/100 令a1a2...an=A (a1-1)*[A/(A-1)]=100/99 a1=100(A-1)/99A+1=(199A-100)/99A,为正整数,设为N (199A-100)/99A=N A=100/(199-99N) N只能为1,2,N=1时,A=1 又a0=1,因此a1=a2=...an=1,代入,不符合题意.N=2时,A=100 a1=2 可以解得a2=5,a3=51 a1a2a3=510>100,不符合题意,无解.