数列{ an}{ bn}满足关系式bn=1*a1+2*a2+3*a3…+nan/1+2+3+…+n,若{bn}为等差数列,求证数列{an}也是等差数
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![数列{ an}{ bn}满足关系式bn=1*a1+2*a2+3*a3…+nan/1+2+3+…+n,若{bn}为等差数列,求证数列{an}也是等差数](/uploads/image/z/3618704-56-4.jpg?t=%E6%95%B0%E5%88%97%7B+an%7D%7B+bn%7D%E6%BB%A1%E8%B6%B3%E5%85%B3%E7%B3%BB%E5%BC%8Fbn%3D1%2Aa1%2B2%2Aa2%2B3%2Aa3%E2%80%A6%2Bnan%2F1%2B2%2B3%2B%E2%80%A6%2Bn%2C%E8%8B%A5%7Bbn%7D%E4%B8%BA%E7%AD%89%E5%B7%AE%E6%95%B0%E5%88%97%2C%E6%B1%82%E8%AF%81%E6%95%B0%E5%88%97%7Ban%7D%E4%B9%9F%E6%98%AF%E7%AD%89%E5%B7%AE%E6%95%B0)
数列{ an}{ bn}满足关系式bn=1*a1+2*a2+3*a3…+nan/1+2+3+…+n,若{bn}为等差数列,求证数列{an}也是等差数
数列{ an}{ bn}满足关系式bn=1*a1+2*a2+3*a3…+nan/1+2+3+…+n,若{bn}为等差数列,求证数列{an}也是等差数
数列{ an}{ bn}满足关系式bn=1*a1+2*a2+3*a3…+nan/1+2+3+…+n,若{bn}为等差数列,求证数列{an}也是等差数
证:
设{bn}公差为d (d为常数).
a1/1=b1 a1=b1
bn=(a1+2a2+3a3+...+nan)/(1+2+3+...+n)
a1+2a2+3a3+...+nan=[n(n+1)/2]bn (1)
a1+2a2+3a3+...+(n+1)a(n+1)=[(n+1)(n+2)/2]b(n+1) (2)
(2)-(1)
(n+1)a(n+1)=[(n+1)(n+2)/2]b(n+1)-[n(n+1)/2]bn
2a(n+1)=(n+2)b(n+1)-nbn
=(n+2)(b1+nd)-n[b1+(n-1)d]
=nb1+n²d+2b1+2nd-nb1-n²d+nd
=3dn+2b1
a(n+1)=(3d/2)n+b1
a(n+1)-an=(3d/2)n+b1-(3d/2)(n-1)-b1=(3/2)d,为定值.
数列{an}是以b1为首项,(3/2)d为公差的等差数列.
设bn = xn + y
则
an = { n(n-1)/2*(xn+y) - n(n+1)/2*[x(n-1)+y] }/n
= (n-1)/2*(xn+y) - (n+1)/2*[x(n-1)+y]
= 1/2 * [ -nx + (x-2y) ]
因此,an也为等差数列