已知数列{An}的前n项和公式Sn=32n-n^2,求新数列{/An/}的n项和Tn
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![已知数列{An}的前n项和公式Sn=32n-n^2,求新数列{/An/}的n项和Tn](/uploads/image/z/677217-57-7.jpg?t=%E5%B7%B2%E7%9F%A5%E6%95%B0%E5%88%97%7BAn%7D%E7%9A%84%E5%89%8Dn%E9%A1%B9%E5%92%8C%E5%85%AC%E5%BC%8FSn%3D32n-n%5E2%2C%E6%B1%82%E6%96%B0%E6%95%B0%E5%88%97%7B%2FAn%2F%7D%E7%9A%84n%E9%A1%B9%E5%92%8CTn)
已知数列{An}的前n项和公式Sn=32n-n^2,求新数列{/An/}的n项和Tn
已知数列{An}的前n项和公式Sn=32n-n^2,求新数列{/An/}的n项和Tn
已知数列{An}的前n项和公式Sn=32n-n^2,求新数列{/An/}的n项和Tn
an=sn-sn-1
=32n-n^2-32n+32+n^2-2n+1
=-2n+33
-2n+33>0 n<33/2≈16
即|an|前十六项和为
31+29+27+25+……+1=256
当n>16时,Tn =(1+2n-33)*(n-16)/2=n^2-32n+264
当n<=16时,Tn =(31+33-2n)*n/2=-n^2+32n
An=Sn-S(n-1)
=32n-n^2-32n+32+n^2-2n+1
=-2n+33
-2n+33>0 n<33/2,n≤16 。
所以
n≤16时,
|An|=|-2n+33 |=-2n+33
Tn =|32n-n^2|=-n^2+32n
当n>16时,
|An|=|-2n+33 |=2n-33
Tn ...
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An=Sn-S(n-1)
=32n-n^2-32n+32+n^2-2n+1
=-2n+33
-2n+33>0 n<33/2,n≤16 。
所以
n≤16时,
|An|=|-2n+33 |=-2n+33
Tn =|32n-n^2|=-n^2+32n
当n>16时,
|An|=|-2n+33 |=2n-33
Tn =|Sn-S16|+|S16|
=|32n-n^2-32*16+16^2|+32*16-16^2
=|32n-n^2-16^2|+16^2
=|-(n-16)^2|+16^2
=(n-16)^2+16^2
=n^2-32n+512
当n≤16时,Tn =-n^2+32n
当n>16时,Tn =n^2-32n+512 。
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