an=d1+d2+d3+...+d2n数列bn,b1=2,bn的m次方=bm的n次方已知n∈N,数列{dn}满足dn=[3+(-1)的n次方]/2,数列{an}满足an=d1+d2+d3+...d2n,数列{bn}中,b1=2,且对任意正整数m,n,(bn)^m=(bm)^a(1)求数列{an}和数列{bn}的通项公式
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![an=d1+d2+d3+...+d2n数列bn,b1=2,bn的m次方=bm的n次方已知n∈N,数列{dn}满足dn=[3+(-1)的n次方]/2,数列{an}满足an=d1+d2+d3+...d2n,数列{bn}中,b1=2,且对任意正整数m,n,(bn)^m=(bm)^a(1)求数列{an}和数列{bn}的通项公式](/uploads/image/z/8633140-52-0.jpg?t=an%3Dd1%2Bd2%2Bd3%2B...%2Bd2n%E6%95%B0%E5%88%97bn%2Cb1%3D2%2Cbn%E7%9A%84m%E6%AC%A1%E6%96%B9%3Dbm%E7%9A%84n%E6%AC%A1%E6%96%B9%E5%B7%B2%E7%9F%A5n%E2%88%88N%2C%E6%95%B0%E5%88%97%7Bdn%7D%E6%BB%A1%E8%B6%B3dn%3D%5B3%2B%28-1%29%E7%9A%84n%E6%AC%A1%E6%96%B9%5D%2F2%2C%E6%95%B0%E5%88%97%7Ban%7D%E6%BB%A1%E8%B6%B3an%3Dd1%2Bd2%2Bd3%2B...d2n%2C%E6%95%B0%E5%88%97%7Bbn%7D%E4%B8%AD%2Cb1%3D2%2C%E4%B8%94%E5%AF%B9%E4%BB%BB%E6%84%8F%E6%AD%A3%E6%95%B4%E6%95%B0m%2Cn%2C%28bn%29%5Em%3D%28bm%29%5Ea%EF%BC%881%EF%BC%89%E6%B1%82%E6%95%B0%E5%88%97%7Ban%7D%E5%92%8C%E6%95%B0%E5%88%97%7Bbn%7D%E7%9A%84%E9%80%9A%E9%A1%B9%E5%85%AC%E5%BC%8F)
an=d1+d2+d3+...+d2n数列bn,b1=2,bn的m次方=bm的n次方已知n∈N,数列{dn}满足dn=[3+(-1)的n次方]/2,数列{an}满足an=d1+d2+d3+...d2n,数列{bn}中,b1=2,且对任意正整数m,n,(bn)^m=(bm)^a(1)求数列{an}和数列{bn}的通项公式
an=d1+d2+d3+...+d2n数列bn,b1=2,bn的m次方=bm的n次方
已知n∈N,数列{dn}满足dn=[3+(-1)的n次方]/2,数列{an}满足an=d1+d2+d3+...d2n,数列{bn}中,b1=2,且对任意正整数m,n,(bn)^m=(bm)^a
(1)求数列{an}和数列{bn}的通项公式2)数列{bn}中的第a1项,第a2项,第a3项.第an项.删去,剩余按从小到大的顺序排成新数列{cn},求数列{cn}的前2n和
an=d1+d2+d3+...+d2n数列bn,b1=2,bn的m次方=bm的n次方已知n∈N,数列{dn}满足dn=[3+(-1)的n次方]/2,数列{an}满足an=d1+d2+d3+...d2n,数列{bn}中,b1=2,且对任意正整数m,n,(bn)^m=(bm)^a(1)求数列{an}和数列{bn}的通项公式
应该是(bn)^m=(bm)^n吧.
(1)
an=d1+d2+...+d(2n)
=(3/2)×2n +(1/2)×(-1)×[1-(-1)^(2n)]/[1-(-1)]
=3n
b1=2>0,又m、n均为正整数,因此bn>0
(bn)^m=(bm)^n
mlg(bn)=nlg(bm)
lg(bn)/n=lg(bm)/m
lg(b1) /1=lg(2)/1=lg2
数列{lg(bn)/n}是各项均为lg2的常数数列.
lg(bn)/n=lg2
bn=2ⁿ
n=1时,b1=2,同样满足通项公式.
综上,数列{an}的通项公式为an=3n;数列{bn}的通项公式为bn=2ⁿ.
(2)
去掉的项为:3、6、9、……项.即每隔两项去掉一项.{cn}的前2n项和={bn}的前3n项和-{bn}中项数是3的整数倍的项的和.
b(3n+3)/b(3n)=2^(3n+3)/2^(3n)=8,为定值.
b3=2³=8,{bn}中项数是3的整数倍的项是以8为首项,8为公比的等比数列,共n项.
T(2n)=c1+c2+...+cn
=2×[2^(3n) -1]/(2-1) -8×(8ⁿ-1)/(8-1)
=(6×8ⁿ+8)/7