已知首项都为1的俩个数列,an,bn,满足anbn+1–an+1bn+2bn+1bn=0,令cn=an/bn

Question

已知首项都为1的俩个数列,an,bn,满足anbn+1–an+1bn+2bn+1bn=0,令cn=an/bn

Answer 1

anb(n+1)-a(n+1)bn+2b(n+1)bn=0
等式两边同除以b(n+1)bn
an/bn -a(n+1)/b(n+1) +2=0
a(n+1)/b(n+1)- an/bn=2,为定值
a1/b1=1/1=1,数列{an/bn}是以1为首项,2为公差的等差数列
cn=an/bn,数列{cn}是以1为首项,2为公差的等差数列
cn=1+2(n-1)=2n-1
数列{cn}的通项公式为cn=2n-1