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,将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
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![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,将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](/uploads/image/z/10343734-70-4.jpg?t=%E8%A7%82%E5%AF%9F%E4%B8%8B%E5%88%97%E7%AD%89%E5%BC%8F%EF%BC%9A1%2F1%2A2%3D1-1%2F2%2C1%2F2%2A3%3D1%2F2-1%2F3%2C1%2F3%2A4%3D1%2F3-1%2F4%2C%E5%B0%86%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%2F1%2A2%2B1%2F2%2A3%2B1%2F3%2A4%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%3A1%2F1%2A2%2B1%2F2%2A3%2B1%2F3%2A4%2B.%2B1%2Fn%28n%2B1%29%3D+%282%29%E7%8C%9C%E6%83%B3%E5%B9%B6%E5%86%99)
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,将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
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,将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
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,将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
这是分式相消的思想!
观察结果:
注意:
1/1*2 = (2-1)/1*2 = 2/1*2 - 1/1*2 = 1/1 - 1/2
1/2*3 = (3-2)/2*3 = 3/2*3 - 2/2*3 = 1/2 - 1/3
1/3*4 = (4-3)/3*4 = 4/3*4 - 3/3*4 = 1/3 - 1/4
上述各式相加:
左边=1/1*2+1/2*3+1/3*4
右边=1/1 - 1/2 + 1/2 - 1/3 + 1/3 - 1/4 = 1- 1/4 = 3/4
因此:
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)* [(n+2)-n]/n(n+2) = (1/2)* [1/n - 1/(n+2)]
3)
1/x(x+3) = (1/3)*[(x+3)-x]/x(x+3)=(1/3)*[1/x - 1(x+3)]
1/(x+3)(x+6) =(1/3)*[(x+6)-(x+3)]/(x+3)(x+6)=(1/3)*[1/(x+3) - 1(x+6)]
1/(x+6)(x+9) =(1/3)*[(x+3)-(x+6)]/(x+6)(x+9)=(1/3)*[1/(x+6) - 1(x+9)]
所以:
1/x(x+3)+1/(x+3)(x+6)+1/(x+6)(x+9)
=(1/3)*[1/x - 1(x+3) + 1/(x+3) - 1(x+6) + 1/(x+6) - 1(x+9)]
=(1/3)*[1/x- 1(x+9)]
=(1/3)*[(x+9-x)/x(x+9)]
=3/x(x+9)
=3/(2x+18) (你写的不清楚,就这么理解了!)
=3/2(x+9)
原式化简成:
3/x(x+9) = 3/2(x+9)
1/x = 1/2
所以:
x=2