Question

观察下列各式，1/1*2=1-1/2 1/2*3=1/2-1/3，1/3*4=1/3-1/4

（1）计算1/1*2+1/2*3+1/3*4+1/4*5+1/5*6=
（2）探究1/1*2+1/2*3+1/3*4+...+1/n（n+1）= (用含有n的式子表示)
（3）若1/1*3+1/3*5+1/5*7+...+1/(2n-1)(2n+1)的值为17/35，求n的值

Answer 1

1/1*2+1/2*3+1/3*4+1/4*5+1/5*6
=1-1/2+1/2-1/3+1/3-1/4+..._1/5-1/6
=1-1/6
=5/6

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

1/1*3+1/3*5+1/5*7+...+1/(2n-1)(2n+1)的值为17/35
(1/2)[1-1/3+1/3-1/5+1/5-1/7+...+1/(2n-1)-1/(2n+1)]=17/35
1-1/(2n+1)=34/35
2n+1=35
n=17

Answer 2

1/1*2+1/2*3+1/3*4+1/4*5+1/5*6
=1-1/2+1/2-1/3+1/3-1/4+..._1/5-1/6
=1-1/6
=5/6

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

1/1*3+1/3*5+1/5*7+...+1/(2n-1)(2n+1)的值为17/35
(1/2)[1-1/3+1/3-1/5+1/5-1/7+...+1/(2n-1)-1/(2n+1)]=17/35
1-1/(2n+1)=34/35
2n+1=35
n=17