数列{bn}满足 3bn+1 + 3bn-1 = bn,b1 =1,求{bn}的通项公式

1个回答

  • 3b(n+1) -bn+ 3b(n-1) = 0

    设3[b(n+1)-xbn] -y[bn-xb(n-1)] = 0

    3b(n+1)-(3x+y)bn+xyb(n-1) = 0

    3x+y=1,xy=3

    x1=(1+i√11)/2,y1=-(1+i3√11)/2

    x2=(1-i√11)/2,y2=-(1-i3√11)/2

    3[b(n+1)-x1bn]=y1[bn-x1b(n-1)]

    3[b(n+1)-x2bn]=y2[bn-x2b(n-1)]

    相除:

    [b(n+1)-x1bn]/[b(n+1)-x2bn]=(y1/y2)[bn-x1b(n-1)]/[bn-x2b(n-1)]

    设[b(n+1)-x1bn]/[b(n+1)-x2bn]=an,a1=[b2-x1b1]/[b2-x2b1]=[b2-x1]/[b2-x2]

    上式变成:

    an=(y1/y2)a(n-1)

    an=a1(y1/y2)^(n-1)

    [b(n+1)-x1bn]/[b(n+1)-x2bn]=an=a1(y1/y2)^(n-1)

    b(n+1)-x1bn=[b(n+1)-x2bn]a1(y1/y2)^(n-1)=[b(n+1)a1(y1/y2)^(n-1)-x2bna1(y1/y2)^(n-1)

    {1-a1(y1/y2)^(n-1)}b(n+1)=[x1-x2a1(y1/y2)^(n-1)]bn

    再化简可将左边(y1/y2)^(n-1)化成(y1/y2)^n,再用叠乘法可求出;

    还少一个条件b2=?,求a1需要.