关于等差数列前N项和的性质的疑惑(1)等差数列an依次每K项之和仍成等差数列,其公差为原公差的K平方倍.(2)若等差数列的项数为2n,则S2n=n(an+a(n+1))(其中an,a(n+1)为中间两项)且S偶-
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![关于等差数列前N项和的性质的疑惑(1)等差数列an依次每K项之和仍成等差数列,其公差为原公差的K平方倍.(2)若等差数列的项数为2n,则S2n=n(an+a(n+1))(其中an,a(n+1)为中间两项)且S偶-](/uploads/image/z/5453003-11-3.jpg?t=%E5%85%B3%E4%BA%8E%E7%AD%89%E5%B7%AE%E6%95%B0%E5%88%97%E5%89%8DN%E9%A1%B9%E5%92%8C%E7%9A%84%E6%80%A7%E8%B4%A8%E7%9A%84%E7%96%91%E6%83%91%EF%BC%881%EF%BC%89%E7%AD%89%E5%B7%AE%E6%95%B0%E5%88%97an%E4%BE%9D%E6%AC%A1%E6%AF%8FK%E9%A1%B9%E4%B9%8B%E5%92%8C%E4%BB%8D%E6%88%90%E7%AD%89%E5%B7%AE%E6%95%B0%E5%88%97%2C%E5%85%B6%E5%85%AC%E5%B7%AE%E4%B8%BA%E5%8E%9F%E5%85%AC%E5%B7%AE%E7%9A%84K%E5%B9%B3%E6%96%B9%E5%80%8D.%EF%BC%882%EF%BC%89%E8%8B%A5%E7%AD%89%E5%B7%AE%E6%95%B0%E5%88%97%E7%9A%84%E9%A1%B9%E6%95%B0%E4%B8%BA2n%2C%E5%88%99S2n%EF%BC%9Dn%28an%EF%BC%8Ba%28n%EF%BC%8B1%29%29%EF%BC%88%E5%85%B6%E4%B8%ADan%2Ca%28n%EF%BC%8B1%29%E4%B8%BA%E4%B8%AD%E9%97%B4%E4%B8%A4%E9%A1%B9%EF%BC%89%E4%B8%94S%E5%81%B6%EF%BC%8D)
关于等差数列前N项和的性质的疑惑(1)等差数列an依次每K项之和仍成等差数列,其公差为原公差的K平方倍.(2)若等差数列的项数为2n,则S2n=n(an+a(n+1))(其中an,a(n+1)为中间两项)且S偶-
关于等差数列前N项和的性质的疑惑
(1)等差数列an依次每K项之和仍成等差数列,其公差为原公差的K平方倍.(2)若等差数列的项数为2n,则S2n=n(an+a(n+1))(其中an,a(n+1)为中间两项)且S偶-S奇=nd,S奇比S偶=an比a(n+1)(3),数列Sn比n是等差数列,公差为二分之d.(4),数列Sn,S(2n)-Sn,S(3n)-S(2n)也成等差数列.
麻烦各位了,
关于等差数列前N项和的性质的疑惑(1)等差数列an依次每K项之和仍成等差数列,其公差为原公差的K平方倍.(2)若等差数列的项数为2n,则S2n=n(an+a(n+1))(其中an,a(n+1)为中间两项)且S偶-
(1),每k项:am+ a(m+1) + ……+a(m-1+k)
a(m+k)+ a(m+1+k)+ ……+a(m-1+2k)
第二行的每项都比第一行多a(m+k)-am=a(m+1+k)-a(m+1)= ……=a(m-1+2k)-a(m-1+k)=k*k=K平方倍
(2)求和公式,S2n=1/2 *2n(an+a(n+1))=n(an+a(n+1))
S偶-S奇
=(a2+a4+a6+……+a2n)-(a1+a3+a5+……+a2n-1)
=(a2-a1)+(a4-a3)+……+(a2n-a2n-1)
=nd
偶数项成等差数列
S偶=a2+a4+a6+……+a2n=1/2 *n(a2+a2n)=na(n+1)
S奇=a1+a3+a5+……+a2n-1=1/2 *n(a1+a2n-1)=nan
S奇比S偶=an比a(n+1)
(3)有公式Sn=na1+n(n-1)d/2
Sn/n=a1+(n-1)d/2
=n*d/2+a1-d/2
成等差数列,公差d/2
(4)Sn =a1+ a2 +……+ an
S2n-Sn=a(n+1)+ a(n+2)+……+ a2n
S3n-S2n=a(2n+1)+a(2n+2)+…… +a3n
每一行的对应项都比上一行多nd
所以(S2n-Sn)-Sn=(S3n-S2n)-(S2n-Sn)=n^2 *d
成等差数列.