数列{an}中,a1=-13,a(n+1)=(2an +3)/an,求{an}的通项公式.注:等号左边的括号表示角标
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![数列{an}中,a1=-13,a(n+1)=(2an +3)/an,求{an}的通项公式.注:等号左边的括号表示角标](/uploads/image/z/2110609-1-9.jpg?t=%E6%95%B0%E5%88%97%7Ban%7D%E4%B8%AD%2Ca1%3D-13%2Ca%EF%BC%88n%2B1%EF%BC%89%3D%EF%BC%882an+%2B3%EF%BC%89%2Fan%2C%E6%B1%82%7Ban%7D%E7%9A%84%E9%80%9A%E9%A1%B9%E5%85%AC%E5%BC%8F.%E6%B3%A8%3A%E7%AD%89%E5%8F%B7%E5%B7%A6%E8%BE%B9%E7%9A%84%E6%8B%AC%E5%8F%B7%E8%A1%A8%E7%A4%BA%E8%A7%92%E6%A0%87)
数列{an}中,a1=-13,a(n+1)=(2an +3)/an,求{an}的通项公式.注:等号左边的括号表示角标
数列{an}中,a1=-13,a(n+1)=(2an +3)/an,求{an}的通项公式.注:等号左边的括号表示角标
数列{an}中,a1=-13,a(n+1)=(2an +3)/an,求{an}的通项公式.注:等号左边的括号表示角标
a(n+1)=(2an+3)/an
a(n+1)+1=(3an+3)/an=3(an +1)/an (1)
a(n+1) -3=(2an+3-3an)/an=(-an +3)/an=-(an -3)/an (2)
(1)/(2)
[a(n+1)+1]/[a(n+1)-3]=(-3)(an +1)/(an -3)
{[a(n+1)+1]/a(n+1)-3}/[(an +1)/(an-3)]=-3,为定值.
(a1+1)/(a1-3)=(-13+1)/(-13-3)=3/4
数列{(an +1)/(an -3)}是以3/4为首项,-3为公比的等比数列.
(an +1)/(an -3)=(3/4)×(-3)^(n-1)=-(-3)ⁿ/4
-(-3)ⁿ×(an-3)=4an+4
an=3 -16/[(-3)ⁿ+4]
n=1时,an=3- 16/(-3+4)=3-16=-13同样满足.
数列{an}的通项公式为an=3 -16/[(-3)ⁿ+4].
a(n+1)=(2an+3)/an
1+a(n+1)=1+(2an+3)/an
=(3an+3)/an
所以1/[1+a(n+1)]=an/(3+3an)
=1/3-1/3*[1/(1+an)]
那么1/[1+a(n+1)]-1/4=-1/3*[1/(1+an)-1/4]
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a(n+1)=(2an+3)/an
1+a(n+1)=1+(2an+3)/an
=(3an+3)/an
所以1/[1+a(n+1)]=an/(3+3an)
=1/3-1/3*[1/(1+an)]
那么1/[1+a(n+1)]-1/4=-1/3*[1/(1+an)-1/4]
而1/(1+a1)-1/4=1/(1-13)-1/4=-1/12-1/4=-1/3≠0
所以数列{1/(1+an)-1/4}是以-1/3为首项、-1/3为公比的等比数列
1/(1+an)-1/4=(-1/3)^n,所以an=-[(-3)^(n+1)+4]/[(-3)^n+4] (n∈N+)
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a(n+1)=(2an+3)/an
a(n+1)+1=(3an+3)/an=3(an +1)/an (1)
a(n+1) -3=(2an+3-3an)/an=(-an +3)/an=-(an -3)/an (2)
(1)/(2)
[a(n+1)+1]/[a(n+1)-3]=(-3)(an +1)/(an...
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a(n+1)=(2an+3)/an
a(n+1)+1=(3an+3)/an=3(an +1)/an (1)
a(n+1) -3=(2an+3-3an)/an=(-an +3)/an=-(an -3)/an (2)
(1)/(2)
[a(n+1)+1]/[a(n+1)-3]=(-3)(an +1)/(an -3)
{[a(n+1)+1]/a(n+1)-3}/[(an +1)/(an-3)]=-3,为定值。
(a1+1)/(a1-3)=(-13+1)/(-13-3)=3/4
数列{(an +1)/(an -3)}是以3/4为首项,-3为公比的等比数列。
(an +1)/(an -3)=(3/4)×(-3)^(n-1)=-(-3)ⁿ/4
-(-3)ⁿ×(an-3)=4an+4
an=3 -16/[(-3)ⁿ+4]
n=1时,an=3- 16/(-3+4)=3-16=-13同样满足。
数列{an}的通项公式为an=3 -16/[(-3)ⁿ+4]。
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