已知a1=2,点（an,a(n+1)）在函数f(x)=x^2+2x的图像上,其中n=1,2,...,
(2）记bn=(1/an)+[1/a（n+2）],求数列{bn}的前n项和Sn.
已知a1=2,点（an,a(n+1)）在函数f(x)=x^2+2x的图像上,其中n=1,2,...,
f(a1)=(a1)^2+2(a1)=f(2)=2^2+2*2=8=(a1)*(a1+2)=a2,
f(a2)=(a2)^2+2(a2)=f(8)=64+16=80=(a2)*(a2+2)=a3,
f(a3)=(a3)^2+2(a3)=f(80)=6400+160=6560=80*(80+2)==(a3)*(a3+2)=a4,
f(a4)=(a4)^2+2(a4)=f(6560)=6560*(6560+2)=(a4)*(a4+2)=a5,
.,
f(a(n-1))=(a(n-1))^2+2(a(n-1))=an,
f(an)=(an)^2+2(an)=a(n+1),
f(a(n+1))=(a(n+1))^2+2(a(n+1))=a(n+2).
(2）记bn=(1/an)+[1/a（n+2）],求数列{bn}的前n项和Sn.
b1=(1/a1)+[1/a3]=1/2+1/80=41/80,
b2=(1/a2)+[1/a4]=1/8+1/6560=83/6560,
b3=(1/a3)+[1/a5],
.,
bn=(1/an)+[1/a(n+2)],
数列{bn}的前n项和Sn
Sn=b1+b2+b3+.+bn=
=（1/a1+1/a3)+(1/a2+1/a4)+(1/a3+1/a5)+...+[1/an+1/a(n+2)]=
={1/a1+1/[(a2)*(a2+2)]}+{1/[(a1)*(a1+2)]+1/[(a3)*(a3+2)]}+.+{1/[(a(n-1))^2+2(a(n-1))]+1/[(a(n+1))^2+2(a(n+1))]},
因为 1/[(a1)*(a1+2)]=（1/2）[1/(a1)-1/(a3)],
1/[(a2)*(a2+2)]=（1/2）[1/(a2)-1/(a4)],
1/[(a3)*(a3+2)]=（1/2）[1/(a3)-1/(a5)],
.,
1/[(a(n-1))*(a(n-1)+2)]=（1/2）[1/(a(n-1)-1/(a(n+1))],
1/[(an))*(an+2)]=（1/2）[1/(an)-1/(a(n+2))],
1/[(a(n+1))*(a(n+1)+2)]=（1/2）[1/(a(n+1)-1/(a(n+3))],
所以,
Sn=1/a1+（1/2）[1/(a2)-1/(a4)]+（1/2）[1/(a1)-1/(a3)]+
+（1/2）[1/(a3)-1/(a5)]+.+（1/2）[1/(a(n-1)-1/(a(n+1))]+
+（1/2）[1/(a(n+1)-1/(a(n+1))]=
=（3/2)(1/a1)+(1/2)(1/a2)-(1/2)[1/(a(n+3))]=
=（3/2)(1/2)+(1/2)(1/8)-(1/2)[1/(a(n+3))]=
=13/16-(1/2)[1/(a(n+3))].