Question

（高考）高中数学：怎么由等差数列求和公式证明是等差数列！是Sn=n(a1+an)/2这个，今天考试的题啊！推...

（高考）高中数学：怎么由等差数列求和公式证明是等差数列！是Sn=n(a1+an)/2这个，今天考试的题啊！推了半天做不出来！

Answer 1

an=Sn-Sn-1=n(a1+an)/2-(n-1)(a1+an-1)/2
2an=a1+an+(n-1)(an-an-1)
an=a1+(n-1)(an-an-1)
an+1=a1+(n)(an+1-an)
两式相减
an+1-an=n(an+1-an)-(n-1)(an-an-1)
an=(an+1+an-1)/2
所以是等差数列

Answer 2

a[n]=S[n+1]-S[n]=a[1]/2+(n+1)a[n+1]/2-na[n]/2
整理有
(n-1)a[n+1]=-a1+na[n]
同样有
(n-2)a[n]=-a1+(n-1)a[n-1]
两式相减并整理即有
a[n+1]-a[n]=a[n]-a[n-1]
此为等差数列定义

Answer 3

Sn=n(a1+an)/2
Sn-1=n(a1+an-1)/2
以上两式做差，
得一方程整理一下，就看出来了

Answer 4

a(n+1)=s(n+1)-s(n)
(n-1)a(n+1)-na(n)=-a(1)
a(n+1)/n - a(n)/(n-1) = -a(1)/[n(n-1)] = a(1)[1/n - 1/(n-1)]
...
a(3)/2-a(2)/1 = a(1)[1/2-1]
a(n+1)/n-a(2)=a(1)[1/n-1]
a(n+1)=a(1)+n[a(2)-a(1)]
等差

Answer 5

Sn-Sn-1=an再证明an是等差数列