探索规律题观察下列等式:1/(1*2)=1-1/2,1/(2*3)=1/2-1/3,1/(3*4)=1/3-1/4以上三个等式两边分别相加得:1/(1*2)+1/(2*3)+1/(3*4)=1-1/2+1/2-1/3+1/3-1/4=1-1/4=3/4(1)直接写出下列各式的计算结果
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![探索规律题观察下列等式:1/(1*2)=1-1/2,1/(2*3)=1/2-1/3,1/(3*4)=1/3-1/4以上三个等式两边分别相加得:1/(1*2)+1/(2*3)+1/(3*4)=1-1/2+1/2-1/3+1/3-1/4=1-1/4=3/4(1)直接写出下列各式的计算结果](/uploads/image/z/3907037-29-7.jpg?t=%E6%8E%A2%E7%B4%A2%E8%A7%84%E5%BE%8B%E9%A2%98%E8%A7%82%E5%AF%9F%E4%B8%8B%E5%88%97%E7%AD%89%E5%BC%8F%EF%BC%9A1%2F%EF%BC%881%2A2%EF%BC%89%3D1-1%2F2%2C1%2F%EF%BC%882%2A3%EF%BC%89%3D1%2F2-1%2F3%2C1%2F%EF%BC%883%2A4%EF%BC%89%3D1%2F3-1%2F4%E4%BB%A5%E4%B8%8A%E4%B8%89%E4%B8%AA%E7%AD%89%E5%BC%8F%E4%B8%A4%E8%BE%B9%E5%88%86%E5%88%AB%E7%9B%B8%E5%8A%A0%E5%BE%97%EF%BC%9A1%2F%EF%BC%881%2A2%EF%BC%89%2B1%2F%EF%BC%882%2A3%EF%BC%89%2B1%2F%EF%BC%883%2A4%EF%BC%89%3D1-1%2F2%2B1%2F2-1%2F3%2B1%2F3-1%2F4%3D1-1%2F4%3D3%2F4%EF%BC%881%EF%BC%89%E7%9B%B4%E6%8E%A5%E5%86%99%E5%87%BA%E4%B8%8B%E5%88%97%E5%90%84%E5%BC%8F%E7%9A%84%E8%AE%A1%E7%AE%97%E7%BB%93%E6%9E%9C)
探索规律题观察下列等式:1/(1*2)=1-1/2,1/(2*3)=1/2-1/3,1/(3*4)=1/3-1/4以上三个等式两边分别相加得:1/(1*2)+1/(2*3)+1/(3*4)=1-1/2+1/2-1/3+1/3-1/4=1-1/4=3/4(1)直接写出下列各式的计算结果
探索规律题
观察下列等式:1/(1*2)=1-1/2,1/(2*3)=1/2-1/3,1/(3*4)=1/3-1/4
以上三个等式两边分别相加得:
1/(1*2)+1/(2*3)+1/(3*4)=1-1/2+1/2-1/3+1/3-1/4=1-1/4=3/4
(1)直接写出下列各式的计算结果:
1/(1*2)+1/(2*3)+1/(3*4)+……+1/[n(n+1)]=_______
(2)猜想并写出:1/[n(n+2)]=_______
(3)探究并解方程:
1/[x(x+3)]+1/[(x+3)(x+6)]+1/[(x+6)(x+9)]=3/[2x+18]
探索规律题观察下列等式:1/(1*2)=1-1/2,1/(2*3)=1/2-1/3,1/(3*4)=1/3-1/4以上三个等式两边分别相加得:1/(1*2)+1/(2*3)+1/(3*4)=1-1/2+1/2-1/3+1/3-1/4=1-1/4=3/4(1)直接写出下列各式的计算结果
(1)1-1/(n+1)
(2)1/n-1/(n+2)=2/[n(n+2)]
(3)1/x-1/(x+3)+1/(x+3)-1/(x+6)+1/(x+6)-1/(x+9)=3/[2x+18]
1/x-1/(x+9)=3/[2x+18]
9/[x(x+9)]=3/[2x+18]
9*[2x+18]=3*[x(x+9)]
18x+162=3x^2+27x
3x^2+9x-162=0
(3x-18)(x+9)=0
x=6或x=-9
经检验,x=-9是原方程的增根,舍去.
所以,原方程的根为x=6.
(1)1/(1*2)+1/(2*3)+1/(3*4)+……+1/[n(n+1)]=1-1/(n+1)
(2)1/[n(n+2)]=(1/2)[1/n-1/(n+2)]
(3)1/[x(x+3)]+1/[(x+3)(x+6)]+1/[(x+6)(x+9)]=3/[2x+18]
(1/3)[1/x-1/(x+9)]=3/[2(x+9)]
1/3x-1/[3(x+9)]=3/[2(x+9)]
[2(x+9)]-2x=9x
2x+18-2x=9x
x=2