题目

1.已知x为整数,且2/(x+3)+2/(3-x)+2x+18/x²-9为整数,求所有符合条件的x的值
2.解方程2(x²+1/x²)-3(x+1/x)-1=0
3.试证：对任意的正整数n,有1/1*2*3 + 1/2*3*4 + ...+1/n(n+1)(n+2)

提问时间：2020-07-26

答案

（1） 2/x+3+2/3-x+2x+18/x^2-9
=[2(X-3)-2(X+3)+2X+18]/(X+3)(X-3)
=2(X+3)/(X+3)(X-3)
=2/(X-3)
∴X-3=±1, ±2
∴X=4, 2, 5, 1
符合条件x的值的和为4+2+5+1=12
（2）令a=x+1/x
两边平方
a²=x²+2+1/x²
所以x²+1/x²=a²-2
所以2(a²-2)-3a-1=0
2a²-3a-5=0
(2a-5)(a+1)=0
a=5/2,a=-1
x+1/x=5/2
2x²-5x+2=0
(2x-1)(x-2)=0
x=1/2,x=2
x+1/x=-1
x²+x+1=0
此方程无解
所以x=1/2,x=2
3）1/n(n+1)(n+2)=1/n·[1/(n+1) - 1/(n+2)]=1/n(n+1) - 1/n(n+2) =[1/n-1/(n+1)] - ½[1/n-1-(n+2)]
=½[1/n-2/(n+1)+1/(n+2)].
∴原式=½(1/1-2/2+1/3)+½(1/2-2/3+1/4)+½(1/3-2/4+1/5)+···+½[1/n-2/(n+1)+1/(n+2)]
=½(1/1+1/2+1/3+···+1/n) - 1·[1/2+1/3+1/4+···+1/(n+1)] + ½[1/3+1/4+···+1/(n+2)]
=½[1+1/2+1/(n+1)+1/(n+2)] + 1·[1/3+1/4+···+1/n] - 1·[1/2+1/3+1/4+···+1/(n+1)]
=1/2+1/4 + ½[1/(n+1)+1/(n+2)] - [1/2+1/(n+1)]
=1/4-½[1/(n+1)-1/(n+2)]