已知正数数列an的前n项和为Sn,若an和2的等差中项与Sn和2的等比中项相等1.求an2.令bn=0.5(a(n+1)/an +an/a(n+1))求 b1+b2+.+bn
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![已知正数数列an的前n项和为Sn,若an和2的等差中项与Sn和2的等比中项相等1.求an2.令bn=0.5(a(n+1)/an +an/a(n+1))求 b1+b2+.+bn](/uploads/image/z/3953947-67-7.jpg?t=%E5%B7%B2%E7%9F%A5%E6%AD%A3%E6%95%B0%E6%95%B0%E5%88%97an%E7%9A%84%E5%89%8Dn%E9%A1%B9%E5%92%8C%E4%B8%BASn%2C%E8%8B%A5an%E5%92%8C2%E7%9A%84%E7%AD%89%E5%B7%AE%E4%B8%AD%E9%A1%B9%E4%B8%8ESn%E5%92%8C2%E7%9A%84%E7%AD%89%E6%AF%94%E4%B8%AD%E9%A1%B9%E7%9B%B8%E7%AD%891.%E6%B1%82an2.%E4%BB%A4bn%3D0.5%EF%BC%88a%EF%BC%88n%2B1%EF%BC%89%2Fan+%2Ban%2Fa%EF%BC%88n%2B1%EF%BC%89%EF%BC%89%E6%B1%82+b1%2Bb2%2B.%2Bbn)
已知正数数列an的前n项和为Sn,若an和2的等差中项与Sn和2的等比中项相等1.求an2.令bn=0.5(a(n+1)/an +an/a(n+1))求 b1+b2+.+bn
已知正数数列an的前n项和为Sn,若an和2的等差中项与Sn和2的等比中项相等
1.求an
2.令bn=0.5(a(n+1)/an +an/a(n+1))求 b1+b2+.+bn
已知正数数列an的前n项和为Sn,若an和2的等差中项与Sn和2的等比中项相等1.求an2.令bn=0.5(a(n+1)/an +an/a(n+1))求 b1+b2+.+bn
1、
由题意,得
(a1+2)/2=√(2a1)
整理,得
(a1-2)²=0
a1-2=0 a1=2
(an+2)/2=√(2Sn)
整理,得
8Sn=(an+2)²
8Sn-1=[a(n-1)+2]²
8an=(an+2)²-[a(n-1)+2]²=an²+4an-a(n-1)²-4a(n-1)
an²-a(n-1)²-4an-4a(n-1)=0
[an+(a(n-1)][an-a(n-1)]-4[an+a(n-1)]=0
[an+a(n-1)][an-a(n-1)-4]=0
数列是正数数列,an+a(n-1)>0,要等式成立,只有an-a(n-1)-4=0
an-a(n-1)=4,为定值.
数列{an}是以2为首项,4为公差的等差数列.
an=2+4(n-1)=4n-2
数列{an}的通项公式为an=4n-2
2、
bn=0.5[a(n+1)/an+an/a(n+1)]
=[(4n+2)/(4n-2)+(4n-2)/(4n+2)]/2
=[1+4/(4n-2)+1-4/(4n+2)]/2
=1+2/[(2n-1)(2n+1)]
=1+1/(2n-1)-1/(2n+1)
b1+b2+...+bn
=n+1/1-1/3+1/3-1/5+...+1/(2n-1)-1/(2n+1)
=n+1-1/(2n+1)