求数列 1,3/2,5/2²,7/2³……的前n项和
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求数列 1,3/2,5/2²,7/2³……的前n项和
求数列 1,3/2,5/2²,7/2³……的前n项和
求数列 1,3/2,5/2²,7/2³……的前n项和
sn=1+3*(1/2)+5*(1/2)^2+7*(1/2)^3+……+(2n-1)*(1/2)^(n-1) ① 两边同乘以1/2
1/2*sn= 1*(1/2)+3*(1/2)^2+5*(1/2)^3+……+(2n-3)*(1/2)^(n-1)+(2n-1)*(1/2)^n ②
①-②1/2*sn=1+2[1/2+(1/2)^2+(1/2)^3+…….+(1/2^(n-1)]-
(2n-1)*(1/2)^n
= 1+ 2{1/2[1-(1/2)^(n-1)]}/[1-1/2] - (2n-1)*(1/2)^n
=1+2[1-(1/2)^(n-1)]-(2n-1)*(1/2)^n
=3-(1/2)^(n-2)-(2n-1)*(1/2)^n=3-(1/2)^(n-2)(1+(2n-1)/4) 两边同乘以2
Sn=6-(2n+3)(1/2)^(n-1)
等于(N2+1)/2(n-1)。分子是n的2次方加1,分母是2的n减1次方