设Sn为数列{an}的前n项和,Sn=kn^2+n,n属于N*,其中k是常数(1)求a1及an(2)若对于任意的m属于N*,a m,a 2m,a 4m成等比数列,求k的值
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![设Sn为数列{an}的前n项和,Sn=kn^2+n,n属于N*,其中k是常数(1)求a1及an(2)若对于任意的m属于N*,a m,a 2m,a 4m成等比数列,求k的值](/uploads/image/z/3754929-57-9.jpg?t=%E8%AE%BESn%E4%B8%BA%E6%95%B0%E5%88%97%7Ban%7D%E7%9A%84%E5%89%8Dn%E9%A1%B9%E5%92%8C%2CSn%3Dkn%5E2%2Bn%2Cn%E5%B1%9E%E4%BA%8EN%2A%2C%E5%85%B6%E4%B8%ADk%E6%98%AF%E5%B8%B8%E6%95%B0%281%29%E6%B1%82a1%E5%8F%8Aan%282%29%E8%8B%A5%E5%AF%B9%E4%BA%8E%E4%BB%BB%E6%84%8F%E7%9A%84m%E5%B1%9E%E4%BA%8EN%2A%2Ca+m%2Ca+2m%2Ca+4m%E6%88%90%E7%AD%89%E6%AF%94%E6%95%B0%E5%88%97%2C%E6%B1%82k%E7%9A%84%E5%80%BC)
设Sn为数列{an}的前n项和,Sn=kn^2+n,n属于N*,其中k是常数(1)求a1及an(2)若对于任意的m属于N*,a m,a 2m,a 4m成等比数列,求k的值
设Sn为数列{an}的前n项和,Sn=kn^2+n,n属于N*,其中k是常数
(1)求a1及an(2)若对于任意的m属于N*,a m,a 2m,a 4m成等比数列,求k的值
设Sn为数列{an}的前n项和,Sn=kn^2+n,n属于N*,其中k是常数(1)求a1及an(2)若对于任意的m属于N*,a m,a 2m,a 4m成等比数列,求k的值
S1=a1=k+1
a2=S2-S1=4k+2-(k+1)=3k+1
a m=Sm-S(m-1)=km^2+m-k(m-1)^2-(m-1)=2km-k+1
a 2m=S2m-S(2m-1)=k(2m)^2+2m-k(2m-1)^2-(2m-1)=4km-k+1
a 4m=S4m-S(4m-1)=k(4m)^2+4m-k(4m-1)^2-(4m-1)=8km-k+1
因为 am,a2m,a4m成等比数列
所以 (a2m)^2=am*a4m
(4km-k+1) ^2 =(2km-k+1)(8km-k+1)
16k^2m^2-8km*(k-1)+(k-1)^2=16k^2m^2-10km*(k-1)+(k-1)^2
2km*(k-1)=0
m属于任意正整数,所以k*(k-1)=0
k=0或1
1:S1=k+1
an=S(n)-S(n-1)=2kn-k+1
a1=2k-k+1=k+1=S1
所以an=2kn-k+1
2:因为 am, a2m, a4m成等比数列
所以 (a2m)*(a2m)=am*a4m
(4km-k+1) (4km-k+1) =(2km-k+1)(8km-k+1)
k=0或k=1