1,已知a+1/a=2,求a^2+!/a^2,猜想a^n+1/a^n(n是任意一个自然数)等于几?并证明你的结论.2,计算:1/a-1+1/(a-1)(a-2)+1/(a-2)(a-3)+...+1/(a-2004)(a-2005)
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![1,已知a+1/a=2,求a^2+!/a^2,猜想a^n+1/a^n(n是任意一个自然数)等于几?并证明你的结论.2,计算:1/a-1+1/(a-1)(a-2)+1/(a-2)(a-3)+...+1/(a-2004)(a-2005)](/uploads/image/z/1127106-18-6.jpg?t=1%2C%E5%B7%B2%E7%9F%A5a%2B1%2Fa%3D2%2C%E6%B1%82a%5E2%2B%21%2Fa%5E2%2C%E7%8C%9C%E6%83%B3a%5En%2B1%2Fa%5En%28n%E6%98%AF%E4%BB%BB%E6%84%8F%E4%B8%80%E4%B8%AA%E8%87%AA%E7%84%B6%E6%95%B0%EF%BC%89%E7%AD%89%E4%BA%8E%E5%87%A0%3F%E5%B9%B6%E8%AF%81%E6%98%8E%E4%BD%A0%E7%9A%84%E7%BB%93%E8%AE%BA.2%2C%E8%AE%A1%E7%AE%97%EF%BC%9A1%2Fa-1%2B1%2F%28a-1%29%28a-2%29%2B1%2F%28a-2%29%28a-3%29%2B...%2B1%2F%28a-2004%29%28a-2005%29)
1,已知a+1/a=2,求a^2+!/a^2,猜想a^n+1/a^n(n是任意一个自然数)等于几?并证明你的结论.2,计算:1/a-1+1/(a-1)(a-2)+1/(a-2)(a-3)+...+1/(a-2004)(a-2005)
1,已知a+1/a=2,求a^2+!/a^2,猜想a^n+1/a^n(n是任意一个自然数)等于几?并证明你的结论.
2,计算:1/a-1+1/(a-1)(a-2)+1/(a-2)(a-3)+...+1/(a-2004)(a-2005)
1,已知a+1/a=2,求a^2+!/a^2,猜想a^n+1/a^n(n是任意一个自然数)等于几?并证明你的结论.2,计算:1/a-1+1/(a-1)(a-2)+1/(a-2)(a-3)+...+1/(a-2004)(a-2005)
1.
a+1/a=2
a^2+1/a^2=(a+1/a)^2-2=4-2=2
a^3+1/a^3=(a+1/a)(a^2+1/a^2)-(a+1/a)=2*2-2=2
猜想a^n+1/a^n=2
证明:(数学归纳法)
设n=k时成立,即a^k+1/a^k=2,a^(k-1)+1/a^(k-1)]=2
a^(k+1)+1/a^(k+1)=[a^k+1/a^k](a+1/a)-[a^(k-1)+1/a^(k-1)]
=2*2-2=2
即 n=k+1亦成立
证明a^n+1/a^n=2 成立
2.
1/a-1+1/(a-1)(a-2)+1/(a-2)(a-3)+...+1/(a-2004)(a-2005)
=1/a-1+[1/(a-2)-1/(a-1)]+[1/(a-3)-1/(a-2)]+...+[1/(a-2005)-1/(a-2004)]
=1/(a-2005)
1.a+1/a=2,a^2-2a+1=0,(a-1)^2=0,a=1,a^2+1/a^2=1+1=2,a^n+1/a^n=2
2.1/(a-1)+1/(a-1)(a-2)+1/(a-2)(a-3)+...+1/(a-2004)(a-2005)
=1/(a-1)+1/(a-2)-1/(a-1)+1/(a-3)-1/(a-4)+...+1/(a-2005)-1/(a-2004)
=1/(a-2005)