已知数列{an}满足a1=2,an=2an-1+2(n∈N*,且n≥2)若数列{bn}满足bn=log2(an+2)设Tn是数列{bn/an+2}的前n项和,求证:TN<3/2
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![已知数列{an}满足a1=2,an=2an-1+2(n∈N*,且n≥2)若数列{bn}满足bn=log2(an+2)设Tn是数列{bn/an+2}的前n项和,求证:TN<3/2](/uploads/image/z/1102236-60-6.jpg?t=%E5%B7%B2%E7%9F%A5%E6%95%B0%E5%88%97%7Ban%7D%E6%BB%A1%E8%B6%B3a1%3D2%2Can%3D2an-1%2B2%28n%E2%88%88N%2A%2C%E4%B8%94n%E2%89%A52%29%E8%8B%A5%E6%95%B0%E5%88%97%7Bbn%7D%E6%BB%A1%E8%B6%B3bn%3Dlog2%28an%2B2%29%E8%AE%BETn%E6%98%AF%E6%95%B0%E5%88%97%EF%BD%9Bbn%2Fan%2B2%EF%BD%9D%E7%9A%84%E5%89%8Dn%E9%A1%B9%E5%92%8C%2C%E6%B1%82%E8%AF%81%EF%BC%9ATN%EF%BC%9C3%2F2)
已知数列{an}满足a1=2,an=2an-1+2(n∈N*,且n≥2)若数列{bn}满足bn=log2(an+2)设Tn是数列{bn/an+2}的前n项和,求证:TN<3/2
已知数列{an}满足a1=2,an=2an-1+2(n∈N*,且n≥2)若数列{bn}满足bn=log2(an+2)设Tn是数列{bn/an+2}的前n
项和,求证:TN<3/2
已知数列{an}满足a1=2,an=2an-1+2(n∈N*,且n≥2)若数列{bn}满足bn=log2(an+2)设Tn是数列{bn/an+2}的前n项和,求证:TN<3/2
证:
n≥2时,
an=2a(n-1)+2
an+2=2a(n-1)+4=2[a(n-1)+2]
(an +2)/[a(n-1)+2]=2,为定值.
a1+2=2+2=4
数列{an +2}是以4为首项,2为公比的等比数列.
an +2=4×2^(n-1)=2^(n+1)
bn=log2(an +2)=log2[2^(n+1)]=n+1
bn/(an +2)=(n+1)/2^(n+1)
Tn=b1/(a1+2)+b2/(a2+2)+...+bn/(an+2)
=2/2²+3/2³+4/2⁴+...+(n+1)/2^(n+1)
Tn/2=2/2³+3/2⁴+...+n/2^(n+1)+(n+1)/2^(n+2)
Tn-Tn/2=Tn/2=1/2 +1/2³+...+1/2^(n+1) -(n+1)/2^(n+2)
Tn=1+1/2²+1/2³+...+1/2ⁿ- (n+1)/2^(n+1)
=1/2 +(1/2+1/2²+1/2³+...+1/2ⁿ) -(n+1)/2^(n+1)
=1/2 +(1/2)(1-1/2ⁿ)/(1-1/2) -(n+1)/2^(n+1)
=3/2 -1/2ⁿ -(n+1)/2^(n+1)