函数f(x)=(x-1)²,数列(an)为等差数列,公差为d,数列(bn)为等比数列,公比为q(q≠0,q≠1).设a1=f(d-1),a2=f(d+1),b1=f(g-1),b2=f(g+1).(1)求数列{an} {bn}的通项公式;(2)设对任意自然数n,都有a(n+1)=c1/b1+c2/b2+
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![函数f(x)=(x-1)²,数列(an)为等差数列,公差为d,数列(bn)为等比数列,公比为q(q≠0,q≠1).设a1=f(d-1),a2=f(d+1),b1=f(g-1),b2=f(g+1).(1)求数列{an} {bn}的通项公式;(2)设对任意自然数n,都有a(n+1)=c1/b1+c2/b2+](/uploads/image/z/10445497-25-7.jpg?t=%E5%87%BD%E6%95%B0f%28x%29%3D%28x-1%29%26sup2%3B%2C%E6%95%B0%E5%88%97%EF%BC%88an%29%E4%B8%BA%E7%AD%89%E5%B7%AE%E6%95%B0%E5%88%97%2C%E5%85%AC%E5%B7%AE%E4%B8%BAd%2C%E6%95%B0%E5%88%97%28bn%29%E4%B8%BA%E7%AD%89%E6%AF%94%E6%95%B0%E5%88%97%2C%E5%85%AC%E6%AF%94%E4%B8%BAq%28q%E2%89%A00%2Cq%E2%89%A01%29.%E8%AE%BEa1%3Df%28d-1%29%2Ca2%3Df%28d%2B1%29%2Cb1%3Df%28g-1%29%2Cb2%3Df%28g%2B1%29.%281%29%E6%B1%82%E6%95%B0%E5%88%97%7Ban%7D+%7Bbn%7D%E7%9A%84%E9%80%9A%E9%A1%B9%E5%85%AC%E5%BC%8F%EF%BC%9B%EF%BC%882%EF%BC%89%E8%AE%BE%E5%AF%B9%E4%BB%BB%E6%84%8F%E8%87%AA%E7%84%B6%E6%95%B0n%2C%E9%83%BD%E6%9C%89a%28n%2B1%29%3Dc1%2Fb1%2Bc2%2Fb2%2B)
函数f(x)=(x-1)²,数列(an)为等差数列,公差为d,数列(bn)为等比数列,公比为q(q≠0,q≠1).设a1=f(d-1),a2=f(d+1),b1=f(g-1),b2=f(g+1).(1)求数列{an} {bn}的通项公式;(2)设对任意自然数n,都有a(n+1)=c1/b1+c2/b2+
函数f(x)=(x-1)²,数列(an)为等差数列,公差为d,数列(bn)为等比数列,公比为q(q≠0,q≠1).设a1=f(d-1),a2=f(d+1),b1=f(g-1),b2=f(g+1).
(1)求数列{an} {bn}的通项公式;
(2)设对任意自然数n,都有a(n+1)=c1/b1+c2/b2+…+cn/bn,求c1+c2+c3+…cn.
(注:数列a,b后面都是角标)
函数f(x)=(x-1)²,数列(an)为等差数列,公差为d,数列(bn)为等比数列,公比为q(q≠0,q≠1).设a1=f(d-1),a2=f(d+1),b1=f(g-1),b2=f(g+1).(1)求数列{an} {bn}的通项公式;(2)设对任意自然数n,都有a(n+1)=c1/b1+c2/b2+
(1) d=a2-a1=f(d+1)-f(d-1)=d^2-(d-2)^2=2(2d-2)
d=4d-4
d=4/3
a1=f(d-1)=f(1/3)=(1/3-1)^2=4/9
所以 an=a1+(n-1)d=4/9+(n-1)*4/3=4n/3-8/9
q=b2/b2=f(q+1)/f(q-1)=q^2/(q-2)^2
q=(q-2)^2
q^2-5q+4=0
q=4 或1(舍去)
b1=f(q-1)=f(3)=4
所以 bn=b1*q^(n-1)=4^n
(2)
a(n+1)=c1/b1+c2/b2+…+cn/bn
a(n)=c1/b1+c2/b2+…+c(n-1)/b(n-1)
两式相减,a(n+1)-a(n)=d=cn/bn
cn=bn*d=4^n*4/3=4^(n+1)/3
所以cn是以
c1=4^2/3=16/3为首项,
qc=cn/c(n-1)=4为公比的等比数列
c1+c2+c3+…cn=c1*(1-qc^n)/(1-qc)=16(4^n-1)/9