- 高一数学题求助,大家快来,明早要交~已知数列{an}满足a(n+
- 已知数列{an}满足a(n+1)=(an)^2-n*an+1(n>=1,n属于整数).
1.能否找到一个等差数列满足上述条件?若能,请写出该等差数列的通项,并证明其是等差数列;若不能,请说明理由.
2.试求1/(a1a3)+1/(a2a4)+1/(a3a5)+...+1/[a(n-1)*a(n+1)]的表达式.
- 存在
当an=n+1时，
a=n+2
(an)^2-n*an+1=(n+1)^2-n(n+1)+1=n+2
满足a=(an)^2-n*an+1
且an=n+1是以2为首项，公差为1的等差数列
a-a=n+2-n=2
1/aa=(1/2)[1/a-1/a]
1/(a1a3)+1/(a2a4)+1/(a3a5)+...+1/[a(n-1)*a(n+1)]
=(1/2)[1/a1-1/a3+1/a2-1/a4+1/a3-1/a5+……+1/a-1/a]
=(1/2)[1/a1+1/a2-1/an-1/a]
=(1/2)[1/2+1/3-1/(n+1)-1/(n+2)]
=5/12-(2n+3)/2(n+1)(n+2)