数列求an

1、n = 1，S1 =a1，( 1 + 2/1 )a1 = 2 - a1，a1 = 1/2；Sn = 2 - ( 1 + 2/n )an； Sn-1 = 2 - [ 1 + 2/( n - 1 ) ]an-1；an = Sn - Sn-1 =[ 1 + 2/( n - 1 ) ]an-1 -( 1 + 2/n )anan + ( 1 + 2/n )an =[ 1 + 2/( n - 1 ) ]an-1an( 2n + 2 )/n = an-1( n + 1 )/( n - 1 )an * 2/n =an-1/( n - 1 )an = an-1 * [ n/( n - 1 ) ]/2a2 = a1[2/1]/2 = 1/2；a3 = a2 * [ 3/2 ]/2 = 1/2 * [ 3/2 ]/2 = 3/2^3；a4 = a3 * [4/3]/2 =3/2^3 * [4/3]/2 = 4/2^4；……；通项公式 an = n/2^n 。2、Sn = 2 - ( 1 + 2/n )an = 2 - ( n + 2 )/n * n/2^n = 2 - ( n + 2 )/2^n = 2 - n/2^n - 1/2^(n-1)(1)、gn = n/2^n，则 Sg = g1 + g2 + …… +gn = 1/2 + 2/4 + 3/8 + 4/16 + …… + (n-1)/2^(n-1) + n/2^n2Sg = 1/1 + 2/2 + 3/4 + 4/8 + …… + n/2^(n-1)Sg = 2Sg - Sg= 1/1 + [ 2/2 - 1/2 ] + [ 3/4 - 2/4 ] + [ 4/8 - 3/8 ] + …… + (n-n+1)/2^(n-1) - n/2^n= 1 + 1/2 + 1/4 + …… + 1/2^(n-1) - n/2^n= [ 1 - (1/2)^n ]/[ 1 - 1/2 ] - n/2^n = 2 - 2/2^n - n/2^n = 2 - ( 2 + n )/2^n；(2)、Hn =1/2^(n-1)，Sh = h1 + h2 + …… +hn =1 + 1/2 + 1/4 + …… + 1/2^(n-1)=[ 1 - (1/2)^n ]/[ 1 - 1/2 ] = 2 - 1/2^(n-1)；(3)、Tn = S1 + S2 + …… +Sn= 2n -Sg - Sh = 2n - 2 +( 2 + n )/2^n - 2 +1/2^(n-1)= 2( n - 2 ) + ( 3 + 2n )/2^(n-1) 。