|
高一数学经典试题：两数列{an}{bn}满足bn=(a1+2a2+3a3+...nan)/(1+2+3+..+n)若{bn}是等差数列，求证{an}也是等差数列。 高一数学经典试题解析：bn=(a1+2a2+3a3+...nan)/(1+2+3+..+n) =（a1+2a2+3a3+....nan）/（n+n^2)/2 ∴(n+n^2)bn=2（a1+2a2+3a3+....nan）.......................1 ∴[(n+1)+(n+1)^2]b(n+1)=2[a1+2a2+3a3+....(n+1)a(n+1）].............2 由1式-2式得： (n+n^2)bn-(n^2+3n+2)b(n+1)=-2(n+1)an+1.............3 若{bn}是等差数列设其公差d 则bn=b(n+1)-d...............4 将4式代入3式得 an+1=(2bn+1+nd)/2=bn+1+nd/2.......5 又∵b(n+1)=b1+nd代入5式 得an+1=b1+nd+nd/2=b1+3n/2d 由bn=(a1+2a2+3a3+...nan)/(1+2+3+..+n)知当n=1时b1=a1 ∴an+1=a1+3n/2d ∴an=a1+（n-1）*（2/3)d ∴{an}也是等差数列.公差为(2/3)d (责任编辑：admin) |