2a（n）=a（n）*a（n+1）+1,a1=2 括号里的是下标,求通项

a(n)a(n+1) = 2a(n) - 1,
a(n)a(n+1) - a(n) = a(n)-1,
a(n)[a(n+1)-1] = a(n)-1,
a(n)[a(n+1)-1] -[a(n+1)-1] = [a(n)-1] - [a(n+1)-1] = [a(n)-1][a(n+1)-1],
若a(n+1)=1,则a(n)=1,...,a(1)=1,与a(1)=2矛盾.
因此,a(n)不为1.
[a(n)-1] - [a(n+1)-1] = [a(n)-1][a(n+1)-1],
1/[a(n+1)-1] - 1/[a(n)-1] = 1,
{1/[a(n)-1] }是首项为1/[a(1)-1]=1,公差为1的等差数列.
1/[a(n)-1] = 1 + (n-1) = n,
a(n)-1 = 1/n,
a(n) = 1 + 1/n = (n+1)/n.