(2014•河南)已知{an}是递增的等差数列,a2,a4是方程x2-5x+6=0的根. (1)求{an}的通项公式; (2)求数列{ an 除以2的n次方 }的前n项和
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![(2014•河南)已知{an}是递增的等差数列,a2,a4是方程x2-5x+6=0的根. (1)求{an}的通项公式; (2)求数列{ an 除以2的n次方 }的前n项和](/uploads/image/z/14415040-64-0.jpg?t=%EF%BC%882014%26%238226%3B%E6%B2%B3%E5%8D%97%EF%BC%89%E5%B7%B2%E7%9F%A5%7Ban%7D%E6%98%AF%E9%80%92%E5%A2%9E%E7%9A%84%E7%AD%89%E5%B7%AE%E6%95%B0%E5%88%97%2Ca2%2Ca4%E6%98%AF%E6%96%B9%E7%A8%8Bx2-5x%2B6%3D0%E7%9A%84%E6%A0%B9%EF%BC%8E+%EF%BC%881%EF%BC%89%E6%B1%82%7Ban%7D%E7%9A%84%E9%80%9A%E9%A1%B9%E5%85%AC%E5%BC%8F%EF%BC%9B+%EF%BC%882%EF%BC%89%E6%B1%82%E6%95%B0%E5%88%97%7B+an+%E9%99%A4%E4%BB%A52%E7%9A%84n%E6%AC%A1%E6%96%B9+%7D%E7%9A%84%E5%89%8Dn%E9%A1%B9%E5%92%8C)
(2014•河南)已知{an}是递增的等差数列,a2,a4是方程x2-5x+6=0的根. (1)求{an}的通项公式; (2)求数列{ an 除以2的n次方 }的前n项和
(2014•河南)已知{an}是递增的等差数列,a2,a4是方程x2-5x+6=0的根. (1)求{an}的通项公式
; (2)求数列{ an 除以2的n次方 }的前n项和
(2014•河南)已知{an}是递增的等差数列,a2,a4是方程x2-5x+6=0的根. (1)求{an}的通项公式; (2)求数列{ an 除以2的n次方 }的前n项和
方程x²-5x+6=0的根是2、3
由于{an}是递增的等差数列,所以a2=2 a4=3
设an=a1+(n-1)d
于是有a2=a1+d=2
a4=a1+3d=3
解得a1=3/2 d=1/2
于是an=3/2+(n-1)/2=n/2+1
2、
an/2^n=(n/2+1)/2^n=n/2^(n+1)+1/2^n
设数列{n/2^(n+1)}前n项和为Sn,数列{n/2^n}为Pn,数列{n/2^(n+1)+1/2^n}前n项和为Tn,
则Tn=Sn+Pn
Pn=1/2+1/2²+1/2³+.+1/2^n=1-(1/2)^n
Sn=1/2²+2/2³+3/2⁴+.+n/2^(n+1)
则2Sn=1/2+2/2²+3/2³+4/2⁴+.+n/2^n
上两式错项相减得
2Sn-Sn=1/2+1/2²+1/2³+1/2⁴+.+1/2^n-n/2^(n+1)
即Sn=1-(1/2)^n-n/2^(n+1)=1-(n+2)/2^(n+1)
于是Tn=Sn+Pn=1-(n+2)/2^(n+1)+1-(1/2)^n=2-(n+4)/2^(n+1)