设数列〔an〕满足a1=1,a2=5/3(5分之3),an+2=5/3an+1-2/3an,(n属于N※).(1)令bn=an+1-an(n=1,2''''''),求数列(bn)的通项公式.(2)求数列(an)的前n项和Sn(备注题中的列如:an+2中的n+2为a的底,其它的
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![设数列〔an〕满足a1=1,a2=5/3(5分之3),an+2=5/3an+1-2/3an,(n属于N※).(1)令bn=an+1-an(n=1,2''''''),求数列(bn)的通项公式.(2)求数列(an)的前n项和Sn(备注题中的列如:an+2中的n+2为a的底,其它的](/uploads/image/z/1090495-55-5.jpg?t=%E8%AE%BE%E6%95%B0%E5%88%97%E3%80%94an%E3%80%95%E6%BB%A1%E8%B6%B3a1%3D1%2Ca2%3D5%2F3%285%E5%88%86%E4%B9%8B3%EF%BC%89%2Can%2B2%3D5%2F3an%2B1%EF%BC%8D2%2F3an%2C%28n%E5%B1%9E%E4%BA%8EN%E2%80%BB%EF%BC%89.%EF%BC%881%EF%BC%89%E4%BB%A4bn%3Dan%2B1%EF%BC%8Dan%28n%3D1%2C2%27%27%27%27%27%27%29%2C%E6%B1%82%E6%95%B0%E5%88%97%EF%BC%88bn%EF%BC%89%E7%9A%84%E9%80%9A%E9%A1%B9%E5%85%AC%E5%BC%8F.%EF%BC%882%EF%BC%89%E6%B1%82%E6%95%B0%E5%88%97%EF%BC%88an%EF%BC%89%E7%9A%84%E5%89%8Dn%E9%A1%B9%E5%92%8CSn%28%E5%A4%87%E6%B3%A8%E9%A2%98%E4%B8%AD%E7%9A%84%E5%88%97%E5%A6%82%EF%BC%9Aan%2B2%E4%B8%AD%E7%9A%84n%2B2%E4%B8%BAa%E7%9A%84%E5%BA%95%2C%E5%85%B6%E5%AE%83%E7%9A%84)
设数列〔an〕满足a1=1,a2=5/3(5分之3),an+2=5/3an+1-2/3an,(n属于N※).(1)令bn=an+1-an(n=1,2''''''),求数列(bn)的通项公式.(2)求数列(an)的前n项和Sn(备注题中的列如:an+2中的n+2为a的底,其它的
设数列〔an〕满足a1=1,a2=5/3(5分之3),an+2=5/3an+1-2/3an,(n属于N※).
(1)令bn=an+1-an(n=1,2''''''),求数列(bn)的通项公式.(2)求数列(an)的前n项和Sn
(备注题中的列如:an+2中的n+2为a的底,其它的an+1也如此)
注意有更正哈:那个a2是等于3分之5的哦~
设数列〔an〕满足a1=1,a2=5/3(5分之3),an+2=5/3an+1-2/3an,(n属于N※).(1)令bn=an+1-an(n=1,2''''''),求数列(bn)的通项公式.(2)求数列(an)的前n项和Sn(备注题中的列如:an+2中的n+2为a的底,其它的
(1)
a(n+2)=(5/3)*a(n+1)-(2/3)*a(n)
a(n+2)=(2/3)*a(n+1)+a(n+1)-(5/3)*a(n)
a(n+2)-a(n+1)=(2/3)[a(n+1)-a(n)]
即b(n+1)=(2/3)b(n)
b(n)是等比数列
b1=a2-a1=5/3-1=2/3
首项为2/3 公比为2/3
b(n)=(2/3)^n
n>=1 n是整数
(2)
a(n)-a(n-1)=b(n-1)=(2/3)^(n-1)
a(n-1)-a(n-2)=b(n-2)=(2/3)^(n-2)
.
a(3)-a(2)=b(2)=(2/3)^2
a(2)-a(1)=b(1)=(2/3)^1
各等式相加得
a(n)-a(1)=(2/3)^1+(2/3)^2+.+(2/3)^(n-2)+(2/3)^(n-1)
a(n)=a1+(2/3)*(1-(2/3)^(n-1))/(1-2/3)
=1+2(1-(2/3)^(n-1))
=3-2(2/3)^(n-1)
Sn=a1+a2+...+a(n)
=3-2*1+3-2*(2/3)+...+3-2(2/3)^(n-1)
=3n-2(1+2/3+.+(2/3^(n-1)))
=3n-2(1-(2/3)^n)/(1-2/3)
=3n-6(1-(2/3)^n)
=3n+6(2/3)^n-6
那个a2=5/3是5分之3?不是3分之5?