已知数列{an}的前n项和Sn=an^2+bn+c(n∈N*),写出{an}是等差数列的充要条件加以证明

来源:学生作业帮助网 编辑:作业帮 时间:2024/11/08 05:59:33

已知数列{an}的前n项和Sn=an^2+bn+c(n∈N*),写出{an}是等差数列的充要条件加以证明
已知数列{an}的前n项和Sn=an^2+bn+c(n∈N*),写出{an}是等差数列的充要条件
加以证明

已知数列{an}的前n项和Sn=an^2+bn+c(n∈N*),写出{an}是等差数列的充要条件加以证明
a(1) = S(1) = a + b + c.
n > 1,
a(n) = S(n) - S(n-1) = an^2 + bn + c - a(n-1)^2 - b(n-1) - c
= 2na + b - a
a(n+1) - a(n) = 2a,n = 2,3,...
若{an}是等差数列,
则,
a(2) - a(1) = 4a + b - a - a - b - c = 2a - c = 2a,
c = 0.

c = 0,
则,
a(2) - a(1) = 2a - c = 2a = a(n+1) - a(n),n = 2,3,...
{an}是等差数列.
因此
c = 0是{an}是等差数列的充要条件.