等差数列an的公差和等比数列bn的公比都是d(d≠1),且a1=b1,a4=b4,a10=b10.求实数a1和b1的值.
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等差数列an的公差和等比数列bn的公比都是d(d≠1),且a1=b1,a4=b4,a10=b10.求实数a1和b1的值.
![等差数列an的公差和等比数列bn的公比都是d(d≠1),且a1=b1,a4=b4,a10=b10.求实数a1和b1的值.](/uploads/image/z/8693426-2-6.jpg?t=%E7%AD%89%E5%B7%AE%E6%95%B0%E5%88%97an%E7%9A%84%E5%85%AC%E5%B7%AE%E5%92%8C%E7%AD%89%E6%AF%94%E6%95%B0%E5%88%97bn%E7%9A%84%E5%85%AC%E6%AF%94%E9%83%BD%E6%98%AFd%EF%BC%88d%E2%89%A01%EF%BC%89%2C%E4%B8%94a1%EF%BC%9Db1%2Ca4%EF%BC%9Db4%2Ca10%EF%BC%9Db10.%E6%B1%82%E5%AE%9E%E6%95%B0a1%E5%92%8Cb1%E7%9A%84%E5%80%BC.)
因为an为等差数列,公差为d;bn为等比数列,公比为d
所以a4=a1+3d;b4=b1×d^3;a1+3d=b1×d^3(1)
所以a10=a1+9d;b10=b1×d^9;a1+9d=b1×d^9(2)
将(1)/(2)得:(a1+3d)/(a1+3d)=1/d^6,即:a1=(9d-3d^7)/(d^6-1)
将(2)-(1)得6d=(d^9-d^3)×b1即:b1=6/[d^2×(d^6-1)]
所以a1/b1=(3d^3-d^9)/2
所以a4=a1+3d;b4=b1×d^3;a1+3d=b1×d^3(1)
所以a10=a1+9d;b10=b1×d^9;a1+9d=b1×d^9(2)
将(1)/(2)得:(a1+3d)/(a1+3d)=1/d^6,即:a1=(9d-3d^7)/(d^6-1)
将(2)-(1)得6d=(d^9-d^3)×b1即:b1=6/[d^2×(d^6-1)]
所以a1/b1=(3d^3-d^9)/2
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