Question

已知数列An满足A1=1,An+An-1=（1/2)^2,n≥2,Sn=A1x2+A2x2^2+A3x2^3+……+An

Answer 1

Sn=A1*2+A2*2²+A3*2³+.+An*2^n ①

2Sn=A1*2²+A2*2³+A3*2^4+.+A(n-1)*2^n+An*2^(n+1) ②

①+②得：3Sn=A1*2+(A2+A1)*2²+(A3+A2)*2³+.+(An+A(n-1))*2^n+An*2^(n+1)

则有,3Sn-An*2^(n+1)=A1*2+(A2+A1)*2²+(A3+A2)*2³+.+(An+A(n-1))*2^n ③

又An+A(n-1)=(1/2)² ,A1=1

则,③式=2+(1/2)²[2²+2³+.+2^n]

=2+(1/2)²[2²(1-2^(n-1))/(1-2)]

=2+(1/2)²(2^(n+1)-4]

=2+2^(n-1)-1

=2^(n-1)+1(题中有n≥2)

所以：3Sn-An*2^(n+1)=2^(n-1)+1