Question

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,

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 三个题都要过程

Answer 1

（1）1/1*2+1/2*3+1/3*4+....+1/n(n+1)=
=1-1/2 + 1/2-1/3 + 1/3-1/4 + .... +1/n - 1/(n+1)
=1 - 1/(n+1)
=n/(n+1)

(2)猜想并写出；1/n(n+2)= (1/2) [1/n - 1/(n+2)] （就是1/n - 1/(n+2) 整个再除以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
1/x(x+3)+1/(x+3)(x+6)+1/(x+6)(x+9)
=(1/3)[1/x -1/(x+3)] + (1/3)[1/(x+3) - 1/(x+6)] + (1/3)[1/(x+6) - 1/(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)
所以
(1/3) [ 1/x - 1/(x+9) ] = 3/2x+18 你这里分母是2x+18，分子是3对吧？
1/x - 1/(x+9) = 9/(2x+18) （同时乘以x(2x+18)）
2x+18 - 2x = 9x
9x = 18
x = 2

Answer 2

(1) 1/1*2+1/2*3+1/3*4+....+1/n(n+1)
=1/1-1/2+1/2-1/3+1/3-1/4+……+1/n-1/(n+1)
=1/1-1/(n-1)
=n/(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+3)+1/(x+3)-1/(x+6)+1/(x+6)-1/(x+9)=3/(2x+18)
1/3(1/x-1/(x+9)=3/(2x+18)
1/3·9/x(x+9)=3/(2x+18)
3/x(x+9)=3/[2(x+9)]
3/x=3/2
x=2
经检验x=2是原方程的解

Answer 3

1.n/(n+1)
2.1/2*(1/n-(n+2))
3.1/x(x+3)+1/(x+3)(x+6)+1/(x+6)(x+9)
=1/3*[1/x-1/(x+3)]+1/3*[1/(x+3)-1/(x+6)]+1/3*[1/(x+6)-1/(x+9)]
=1/3*[1/x-1/(x+9)]
又等于3/2x+18
解得x=2