正数列an的前n项和Sn满足：rsn=anan＋1-1,a1=a＞0,常数r∈N,n∈N*

Question

正数列an的前n项和Sn满足：rsn=anan＋1-1,a1=a＞0,常数r∈N,n∈N*

Answer 1

rsn=ana-1,①
n=1时ra1=a1a2-1,a1=a,
∴a2=r+1/a.
n>1时rs=aan-1,②
①-②,ran=an(a-a),
a1=a＞0,常数r∈N,
∴an≠0,
∴a-a=r,
∴a=a1+(m-1)r=a+(m-1)r;
a=a2+(m-1)r=1/a+mr,m∈N*.