有数列{an},a1=5/6,若二次方程a下标(n-1)x^2减a下标(n)x+1=0有根x,y且满足3x-xy+3y=11.求证{an-1/2}为等比数列 2.通向公式an 3.前n项和Sn
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![有数列{an},a1=5/6,若二次方程a下标(n-1)x^2减a下标(n)x+1=0有根x,y且满足3x-xy+3y=11.求证{an-1/2}为等比数列 2.通向公式an 3.前n项和Sn](/uploads/image/z/12687073-25-3.jpg?t=%E6%9C%89%E6%95%B0%E5%88%97%7Ban%7D%2Ca1%3D5%2F6%2C%E8%8B%A5%E4%BA%8C%E6%AC%A1%E6%96%B9%E7%A8%8Ba%E4%B8%8B%E6%A0%87%28n-1%29x%5E2%E5%87%8Fa%E4%B8%8B%E6%A0%87%EF%BC%88n%EF%BC%89x%2B1%3D0%E6%9C%89%E6%A0%B9x%2Cy%E4%B8%94%E6%BB%A1%E8%B6%B33x-xy%2B3y%3D11.%E6%B1%82%E8%AF%81%7Ban-1%2F2%7D%E4%B8%BA%E7%AD%89%E6%AF%94%E6%95%B0%E5%88%97+2.%E9%80%9A%E5%90%91%E5%85%AC%E5%BC%8Fan+3.%E5%89%8Dn%E9%A1%B9%E5%92%8CSn)
有数列{an},a1=5/6,若二次方程a下标(n-1)x^2减a下标(n)x+1=0有根x,y且满足3x-xy+3y=11.求证{an-1/2}为等比数列 2.通向公式an 3.前n项和Sn
有数列{an},a1=5/6,若二次方程a下标(n-1)x^2减a下标(n)x+1=0有根x,y且满足3x-xy+3y=1
1.求证{an-1/2}为等比数列 2.通向公式an 3.前n项和Sn
有数列{an},a1=5/6,若二次方程a下标(n-1)x^2减a下标(n)x+1=0有根x,y且满足3x-xy+3y=11.求证{an-1/2}为等比数列 2.通向公式an 3.前n项和Sn
a(n-1)x²-an x+1=0
两根分别为x、y.由韦达定理得
x+y=an/a(n-1)
xy=1/a(n-1)
3x-xy+3y=1
3(x+y)-xy=1
3an/a(n-1)-1/a(n-1)=1
整理,得
3an=a(n-1)+1
3an-3/2=a(n-1)-1/2
(an- 1/2)/[a(n-1)-1/2]=1/3,为定值.
a1-1/2=5/6-1/2=1/3
数列{an -1/2}是以1/3为首项,1/3为公比的等比数列.
an-1/2=1/3ⁿ
an=1/3ⁿ+1/2
n=1时,a1=1/3+1/2=5/6,同样满足.
数列{an}的通项公式为an=1/3ⁿ+1/2.
Sn=a1+a2+...+an
=1/3+1/2+1/3²+1/2+...+1/3ⁿ+1/2
=(1/3+1/3²+...+1/3ⁿ)+n/2
=(1/3)(1-1/3ⁿ)/(1-1/3)+n/2
=1/(2×3ⁿ) + n/2- 1/2
a(n-1)x²-an x+1=0
两根分别为x、y。由韦达定理得
x+y=an/a(n-1)
xy=1/a(n-1)
3x-xy+3y=1
3(x+y)-xy=1
3an/a(n-1)-1/a(n-1)=1
整理,得
3an=a(n-1)+1
3an-3/2=a(n-1)-1/2
(an- 1/2...
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a(n-1)x²-an x+1=0
两根分别为x、y。由韦达定理得
x+y=an/a(n-1)
xy=1/a(n-1)
3x-xy+3y=1
3(x+y)-xy=1
3an/a(n-1)-1/a(n-1)=1
整理,得
3an=a(n-1)+1
3an-3/2=a(n-1)-1/2
(an- 1/2)/[a(n-1)-1/2]=1/3,为定值。
a1-1/2=5/6-1/2=1/3
数列{an -1/2}是以1/3为首项,1/3为公比的等比数列。
an-1/2=1/3ⁿ
an=1/3ⁿ+1/2
n=1时,a1=1/3+1/2=5/6,同样满足。
数列{an}的通项公式为an=1/3ⁿ+1/2。
Sn=a1+a2+...+an
=1/3+1/2+1/3²+1/2+...+1/3ⁿ+1/2
=(1/3+1/3²+...+1/3ⁿ)+n/2
=(1/3)(1-1/3ⁿ)/(1-1/3)+n/2
=1/(2×3ⁿ) + n/2- 1/2
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