Question

已知数列an中，a1=1，当n≥2时，其前n项和Sn满足Sn^2=an(Sn-1/2)（1）求an

Answer 1

put n=2
(a1+a2)^2 = a2(a1+a2-1/2)
(1+a2)^2=a2(a2+1/2)
2a2+1 = (1/2)a2
(3/2)a2 =-1
a2= -2/3
S2 =a1+a2 = 1/3

(Sn)^2=an(Sn-1/2)
=(Sn-S(n-1)(Sn-1/2)
=(Sn)^2-(1/2)Sn - SnS(n-1) +(1/2)S(n-1)
1 = 1/2(1/Sn - 1/S(n-1) )
1/Sn - 1/S(n-1) =2
1/Sn -1/S2=2(n-2)
1/Sn= 3+2(n-2)
Sn = 1/(2n-1) (1)
S(n-1) = 1/(2n-3) (2)
(1)-(2)
an =1/(2n-1) -1/(2n-3)