数列{an}中,a1=2,且满足点(an,a(n+1))在函数f(x)=x^2+2x的图像上(n∈N*)(1)求证数列{lg(1+an)}是等比数列(2)设Tn=(1+a1)(1+a2)…(1+an),求Tn及数列{an}的通项(3)bn=2*(1/an-1/(a(n+1)))(n∈N*),Sn=b1+b2+…bn
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![数列{an}中,a1=2,且满足点(an,a(n+1))在函数f(x)=x^2+2x的图像上(n∈N*)(1)求证数列{lg(1+an)}是等比数列(2)设Tn=(1+a1)(1+a2)…(1+an),求Tn及数列{an}的通项(3)bn=2*(1/an-1/(a(n+1)))(n∈N*),Sn=b1+b2+…bn](/uploads/image/z/3757081-49-1.jpg?t=%E6%95%B0%E5%88%97%7Ban%7D%E4%B8%AD%2Ca1%3D2%2C%E4%B8%94%E6%BB%A1%E8%B6%B3%E7%82%B9%28an%2Ca%28n%2B1%EF%BC%89%29%E5%9C%A8%E5%87%BD%E6%95%B0f%28x%29%3Dx%5E2%2B2x%E7%9A%84%E5%9B%BE%E5%83%8F%E4%B8%8A%EF%BC%88n%E2%88%88N%2A%29%EF%BC%881%EF%BC%89%E6%B1%82%E8%AF%81%E6%95%B0%E5%88%97%EF%BD%9Blg%281%2Ban%29%7D%E6%98%AF%E7%AD%89%E6%AF%94%E6%95%B0%E5%88%97%EF%BC%882%EF%BC%89%E8%AE%BETn%3D%281%2Ba1%29%281%2Ba2%29%E2%80%A6%281%2Ban%29%2C%E6%B1%82Tn%E5%8F%8A%E6%95%B0%E5%88%97%7Ban%7D%E7%9A%84%E9%80%9A%E9%A1%B9%EF%BC%883%EF%BC%89bn%3D2%2A%281%2Fan-1%2F%28a%28n%2B1%29%29%EF%BC%89%EF%BC%88n%E2%88%88N%2A%29%2CSn%3Db1%2Bb2%2B%E2%80%A6bn)
数列{an}中,a1=2,且满足点(an,a(n+1))在函数f(x)=x^2+2x的图像上(n∈N*)(1)求证数列{lg(1+an)}是等比数列(2)设Tn=(1+a1)(1+a2)…(1+an),求Tn及数列{an}的通项(3)bn=2*(1/an-1/(a(n+1)))(n∈N*),Sn=b1+b2+…bn
数列{an}中,a1=2,且满足点(an,a(n+1))在函数f(x)=x^2+2x的图像上(n∈N*)
(1)求证数列{lg(1+an)}是等比数列
(2)设Tn=(1+a1)(1+a2)…(1+an),求Tn及数列{an}的通项
(3)bn=2*(1/an-1/(a(n+1)))(n∈N*),Sn=b1+b2+…bn(n∈N*),求Sn,并证明:Sn+2/(3*Tn-1)=1
数列{an}中,a1=2,且满足点(an,a(n+1))在函数f(x)=x^2+2x的图像上(n∈N*)(1)求证数列{lg(1+an)}是等比数列(2)设Tn=(1+a1)(1+a2)…(1+an),求Tn及数列{an}的通项(3)bn=2*(1/an-1/(a(n+1)))(n∈N*),Sn=b1+b2+…bn
(1)
因为(an,an+1)在函数f(x)=x^2+2x的图象上
an+1=an^2+2an
1+an+1=an^2+2an +1=(1+an)^2
lg(1+an+1)=2lg(1+an),(n>=1),所以{lg(1+an)}等比
(2)
{lg(1+an)}等比
所以,lg(1+an)=2^(n-1)*lg(1+a1)=lg3*2^(n-1)
lgTn
=lg[(1+a1)(1+a2)…(1+an)]
=lg(1+a1)+lg(1+a2)+…+lg(1+an)
=lg3*[1+2+…+2^(n-1)]=lg3*(1-2^n)/(1-2)=lg3*(2^n-1)
所以Tn=3*10^(2^n-1)
(3)cn=2^n-1;
根据题意:
tn=(2^1+2^2+2^3+.+2^n)-n;
=2(1-2^n)/(1-2)-n
=2^(n+1)-2-n.