Question

已知sn是数列an的前n项和,an>0(n=1,2,3...),且2sn=an+(1/an)(1)求出a1,a2,a3并

Answer 1

an>0(n=1,2,3...),且2sn=an+(1/an),

n=1时,2a1=a1+1/a1,a1^2=1,a1=1.

n=2时2（1+a2)=a2+1/a2,a2^2+2a2-1=0,a2=√2-1.

n=3时2（√2+a3)=a3+1/a3,a3^2+2√2a3-1=0,a3=√3-√2.

假设ak=√k-√（k-1),k∈N+,那么Sk=√k,

n=k+1时2（√k+a)=a+1/a,

∴（a)^2+2√ka-1=0,

a=√（k+1)-√k,

∴对任意n∈N+,都有an=√n-√（n-1).