数列{an}中a1=1 2an+1=(1+1/n)的平方*an 【n属于正整数,2an+1是指a的第n+1项的2倍】(1)令Bn=an/(n的平方) 求证:{bn}等比数列(2)令Cn=an+1-(1/2)*an求数列{an}的前n项和Sn(3)求数列{an}的前n项
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![数列{an}中a1=1 2an+1=(1+1/n)的平方*an 【n属于正整数,2an+1是指a的第n+1项的2倍】(1)令Bn=an/(n的平方) 求证:{bn}等比数列(2)令Cn=an+1-(1/2)*an求数列{an}的前n项和Sn(3)求数列{an}的前n项](/uploads/image/z/1085209-25-9.jpg?t=%E6%95%B0%E5%88%97%7Ban%7D%E4%B8%ADa1%3D1+2an%2B1%3D%EF%BC%881%2B1%2Fn%EF%BC%89%E7%9A%84%E5%B9%B3%E6%96%B9%2Aan+%E3%80%90n%E5%B1%9E%E4%BA%8E%E6%AD%A3%E6%95%B4%E6%95%B0%2C2an%2B1%E6%98%AF%E6%8C%87a%E7%9A%84%E7%AC%ACn%2B1%E9%A1%B9%E7%9A%842%E5%80%8D%E3%80%91%EF%BC%881%EF%BC%89%E4%BB%A4Bn%3Dan%2F%EF%BC%88n%E7%9A%84%E5%B9%B3%E6%96%B9%EF%BC%89+%E6%B1%82%E8%AF%81%EF%BC%9A%7Bbn%7D%E7%AD%89%E6%AF%94%E6%95%B0%E5%88%97%EF%BC%882%EF%BC%89%E4%BB%A4Cn%3Dan%2B1-%EF%BC%881%2F2%EF%BC%89%2Aan%E6%B1%82%E6%95%B0%E5%88%97%7Ban%7D%E7%9A%84%E5%89%8Dn%E9%A1%B9%E5%92%8CSn%EF%BC%883%EF%BC%89%E6%B1%82%E6%95%B0%E5%88%97%7Ban%7D%E7%9A%84%E5%89%8Dn%E9%A1%B9)
数列{an}中a1=1 2an+1=(1+1/n)的平方*an 【n属于正整数,2an+1是指a的第n+1项的2倍】(1)令Bn=an/(n的平方) 求证:{bn}等比数列(2)令Cn=an+1-(1/2)*an求数列{an}的前n项和Sn(3)求数列{an}的前n项
数列{an}中a1=1 2an+1=(1+1/n)的平方*an 【n属于正整数,2an+1是指a的第n+1项的2倍】
(1)令Bn=an/(n的平方) 求证:{bn}等比数列
(2)令Cn=an+1-(1/2)*an求数列{an}的前n项和Sn
(3)求数列{an}的前n项和Tn
数列{an}中a1=1 2an+1=(1+1/n)的平方*an 【n属于正整数,2an+1是指a的第n+1项的2倍】(1)令Bn=an/(n的平方) 求证:{bn}等比数列(2)令Cn=an+1-(1/2)*an求数列{an}的前n项和Sn(3)求数列{an}的前n项
(1)
2a(n+1)=(1+1/n)^2*an=((n+1)^2/n^2)*an
2a(n+1)/(n+1)^2=an/n^2
2b(n+1)=bn
b(n+1)=bn/2
得证
(2)
b1/1^2=1
bn=2^(1-n)
an=n^2*2^(1-n)=n^2/2^(n-1)
Cn=a(n+1)-(1/2)*an=(n+1)^2/2^n-n^2/2^n=(2n+1)/2^n
Sn可用错位相减法求和
(3)
an=n^2/2^(n-1)
S(n-1)=an+(a(n-1)+a(n-2)+……+a2)/2-a1/2
=(an+a(n-1)+a(n-2)+……+a2+a1)/2+an/2-a1
=Tn/2+an/2-a1
Tn=2S(n-1)+2a1-an
带入计算即可……