求和sn=1*2¹+3*2²+5*2³+...+(2n-1)*2ˆn

求和sn=1*2¹+3*2²+5*2³+...+(2n-1)*2ˆn
求和sn=1*2¹+3*2²+5*2³+...+(2n-1)*2ˆn

求和sn=1*2¹+3*2²+5*2³+...+(2n-1)*2ˆn
sn=1*2¹+3*2²+5*2³+...+(2n-1)*2ˆn.(1)
2Sn=1*2^2+3*2^3+5*2^4+.+(2n-3)*2^(n)+(2n-1)*2^(n+1).(2)
(1)-(2)
Sn=2+2[2^2+2^3+...+2^(n)]-(2n-1)*2^(n+1)
=-2+2[2+2^2+2^3+...+2^(n)]-(2n-1)*2^(n+1)
=-2+2[2(1-2^(n))/(1-2)]-(2n-1)*2^(n+1)
=2^(n+2)-(2n-1)*2^(n+1)-4
=2^(n+1)[-2n-1/2]-4

sn=1*2¹+3*2²+5*2³+...+(2n-1)*2ˆn……①
2sn=1*2²+3*2³+5*2^4+...+(2n-1)*2ˆn+1……②
由②-①得sn=-2-（2³+2^4+2^5+……2ˆn+1）+(2n-1)*2ˆn+1
然后用等比数列求和公式即可求出