an=1+2+2²+...+2^(n-1)=1×(2ⁿ-1)/(2-1)=2ⁿ-1
Sn=a1+a2+...+an
=(2+2²+...+2ⁿ)-n
=2×(2ⁿ-1)/(2-1) -n
=2^(n+1)-n-2
Tn=S1+S2+...+Sn
=[2²+2³+...+2^(n+1)] -n²-2n
=4×(2ⁿ-1)/(2-1) -n²-2n
=2^(n+2) -n²-2n -4
Tn>1000
2^(n+2)-n²-2n-4>1000
2^(n+2)>1003+(n+1)²
2^9=512 1003>512
2^99
n>7
n=8时,2^10=1024 1003+(8+1)²=1003+81=1084>1024
n=9时,2^11=2048 1003+(9+1)²=1003+100=1103