设数列An的前n项和为Sn,已知A1=1.A2=6,A3=11,且(5n-8)S(n+1)-(5n+2)Sn=-20n-8,求证An为等差数列
问题描述:
设数列An的前n项和为Sn,已知A1=1.A2=6,A3=11,且(5n-8)S(n+1)-(5n+2)Sn=-20n-8,求证An为等差数列
答
证明:(5n-8)S(n+1)-(5n+2)Sn=-20n-8(5n-8)an+(5n-8)Sn-(5n+2)Sn=-20n-8(5n-8)an-10Sn=-20n-8 ①令n=n-1,得(5n-13)a(n-1)-10S(n-1)=-20n+12 ②①-②,得(5n-18)an-(5n-13)a(n-1)=-20 ③再令n=n-1,得(5n-23)a(n-1)-(...