1.因为a^2-3a+1=0,所以a+1/a=3,
所以(a+1/a)^2=a^2+1/a^2+2=9,
所以a^2+1/a^2=7,
所以a^2/(a^4+1)=1/(a^2+1/a^2)=1/7；
2.[1]原式=(1+3+5+...+17)+(1/2+1/4+...+1/512)
=9^2+1/2*[1-(1/2)^9]/(1-1/2)
=81+511/512
=81 511/512；
[2]因为通项公式是an=(2n-1)/2^n,
所以Sn=a1+a2+...+an
=1/2+3/4+...+(2n-1)/2^n,
所以Sn/2=1/4+3/8+...+(2n-3)/2^n+(2n-1)/2^(n+1),
所以Sn-Sn/2=1/2+2/4+2/8+...+2/2^n-(2n-1)/2^(n+1),
所以Sn/2=1/2+1-1/2^(n-1)-(2n-1)/2^(n+1)
=3/2-(2n+3)/2^(n+1),
所以Sn=3-(2n+3)/2^n,
所以1/2+3/4+5/8+7/16+...+19/1024
=S10=3-23/1024=2 1001/1024；
[3]因为1/(3n-2)(3n+1)=[1/(3n-2)-1/(3n+1)]/3,
所以1/1x4+1/4x7+1/7x10+...+1/(3n-2)x(3n+1)
=[1/1-1/4+1/4-1/7+...+1/(3n-2)-1/(3n+1)]/3
=n/(3n+1)；
[4]因为1x2+2x3+3x4+...+n(n+1)
=(1^2+2^2+...+n^2)+(1+2+...+n)
=n(n+1)(2n+1)/6+n(n+1)/2
=n(n+1)(n+2)/3.