数列{an}中,a3=1,a1+a2+...+an=a(n+1)(n=1,2,3...)

a(1)+a(2)+...+a(n)
=
a(n+1),
a(n+2)
=
a(1)+a(2)+...+a(n)+a(n+1)
=
a(n+1)
+
a(n+1)
=
2a(n+1),
a(1+2)
=
a(3)
=
1
=
2a(1+1)
=
2a(2),
a(2)
=
1/2.
a(3)
=
1
=
a(1)
+
a(2)
=
a(1)
+
1/2,
a(1)
=
1/2.
{a(n+1)}是首项为a(2)=1/2,
公比为2的等比数列。
a(n+1)
=
(1/2)*2^(n-1)
=
2^(n-2),
a(1)=1/2,
n>=2时，a(n)=
2^(n-3).
s(n)
=
a(1)+a(2)+...+a(n)
=
a(n+1)
=
2^(n-2).
b(n)=log_{2}[s(n)]
=
log_{2}[2^(n-2)]
=
n-2,
1
+
n(n+1)(n+2)s(n)
=
1
+
n(n+1)(n+2)2^(n-2)
=
c(n)*b(n+3)b(n+4)
=
(n+1)(n+2)c(n),
c(n)
=
1/[(n+1)(n+2)]
+
n2^(n-2)
=
1/(n+1)
-
1/(n+2)
+
d(n),
d(n)
=
n2^(n-2),
D(n)
=
d(1)+d(2)+d(3)+...+d(n-1)+d(n)
=2^(1-2)
+
2*2^(2-2)
+
3*2^(3-2)
+
...
+
(n-1)2^(n-3)
+
n2^(n-2),
2D(n)
=
2^(2-2)
+
2*2^(3-2)
+
...
+
(n-1)2^(n-2)
+
n2^(n-1),
D(n)
=
2D(n)-D(n)
=
-2^(1-2)
-
2^(2-2)
-
2^(3-2)
-
...
-
2^(n-2)
+
n2^(n-1)
=
n2^(n-1)
-
2^(-1)[1
+
2
+
...
+
2^(n-1)]
=
n2^(n-1)
-
(1/2)[2^n
-
1]/(2-1)
=
n2^(n-1)
-
[2^n
-
1]/2
=
(n-1)2^(n-1)
+
1/2,
t(n)
=
c(1)+c(2)+...+c(n-1)+c(n)
=
[1/2-1/3
+
1/3-1/4
+
...
+
1/n-1/(n+1)
+
1/(n+1)-1/(n+2)]
+
D(n)
=
1/2
-
1/(n+2)
+
(n-1)2^(n-1)
+
1/2
=
1
-
1/(n+2)
+
(n-1)2^(n-1)
=
(n+1)/(n+2)
+
(n-1)2^(n-1)