附加题(10分,总分120以上有效)(1)设函数f(x)=(x-3)3+x-1,{an}是公差不为0的等差数列,f(a1)+f(a2)+…+f(a7)=14,则a1+a2+…+a7=______(2)

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附加题(10分,总分120以上有效)(1)设函数f(x)=(x-3)3+x-1,{an}是公差不为0的等差数列,f(a1)+f(a2)+…+f(a7)=14,则a1+a2+…+a7=______(2)

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附加题(10分,总分120以上有效)
(1)设函数f(x)=(x-3)3+x-1,{an}是公差不为0的等差数列,f(a1)+f(a2)+…+f(a7)=14,则a1+a2+…+a7=______
(2)若Sn=sin
π
7
+sin
7
+…+sin
7
(n∈N+),则在S1,S2,…S100中,正数的个数是______.
题型:解答题难度:中档来源:不详

答案

(1)∵f(x)=(x-3)3+x-1,∴f(x)-2=(x-3)3+x-3,
令g(x)=f(x)-2,
∴g(x)关于(3,0)对称,
∵f(a1)+f(a2)+…+f(a7)=14,
∴f(a1)-2+f(a2)-2+…+f(a7)-2=0
∴g(a1)+g(a2)+…+g(a7)=0,
∴g(a4)为g(x)与x轴的交点,
因为g(x)关于(3,0)对称,所以a4=3,
∴a1+a2+…+a7=7a4=21,
故答案为:21.
(2)∵sinclass="stub"π
7
>0,sinclass="stub"2π
7
>0,…,sinclass="stub"6π
7
>0,sinclass="stub"7π
7
=0,sinclass="stub"8π
7
<0,…,sinclass="stub"13π
7
<0,sinclass="stub"14π
7
=0,
∴S1=sinclass="stub"π
7
>0,
S2=sinclass="stub"π
7
+sinclass="stub"2π
7
>0,…,
S8=sinclass="stub"π
7
+sinclass="stub"2π
7
+…+sinclass="stub"6π
7
+sinclass="stub"7π
7
+sinclass="stub"8π
7
=sinclass="stub"2π
7
+…+sinclass="stub"6π
7
+sinclass="stub"7π
7
>0,
…,
S12>0,
而S13=sinclass="stub"π
7
+sinclass="stub"2π
7
+…+sinclass="stub"6π
7
+sinclass="stub"7π
7
+sinclass="stub"8π
7
+sinclass="stub"9π
7
+…+sinclass="stub"13π
7
=0,
S14=S13+sinclass="stub"14π
7
=0+0=0,
又S15=S14+sinclass="stub"15π
7
=0+sinclass="stub"π
7
=S1>0,S16=S2>0,…S27=S13=0,S28=S14=0,
∴S14n-1=0,S14n=0(n∈N*),在1,2,…100中,能被14整除的共7项,
∴在S1,S2,…,S100中,为0的项共有14项,其余项都为正数.
故在S1,S2,…,S100中,正数的个数是86.
故答案为:86.

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