(1)
a(n+1)=2a2-3^n
n=1,a(2)=2a(2)-3,a(2)=3
a(n+1)=6-3^n
a(n)=6-3^(n-1)
(2)
n为偶数时,
Sn=1-3+5-7+...+(-1)^(n-1)(2n-1)
=1-3+5-7+...+(2n-3)-(2n-1)
=-2-2+...-2
=-2*n/2
=-n
n为奇数时；
Sn=1-3+5-7+...+(-1)^(n-1)(2n-1)
=1-3+5-7+...+(2n-5)-(2n-3)+(2n-1)
=-2*(n-1)/2+(2n-1)
=n
(3)
11...11{共2n个1}-22.22{共n个2}
=[10^(2n-1)+...+10+1]-2*[10^(n-1)+...+10+1]
=[10^(2n)-1]/9-2*(10^n-1)/9
=[10^(2n)-2*10^n+1]/9
=(10^n-1)^2/9
根号(11...11{共2n个1}-22.22{共n个2}）
=(10^n-1)/3
Sn=根号(11-2)+根号(1111-22)+.+根号(11...11{共2n个1}-22.22{共n个2}）
=(10^1-1)/3+(10^2-1)/3+...+(10^n-1)/3
=(10^1+10^2+...+10^n)/3-n/3
=[10^(n+1)-10]/27-n/3
=[10^(n+1)-10-9n]/27
(4)
S(n)=2^n-1
S(n-1)=2^(n-1)-1
a(n)=S(n)-S(n-1)=2^(n-1)
a1^2+a2^2+...+an^2
=2^0+2^2+...+2^(2n-2)
=(4^n-1)/3