已知数列{an}满足2an+1=an+an+2(n∈正整数),它的前n项和为sn,且a3=-6,s6=-30,.求数列{an}的前n项和的最小值
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![已知数列{an}满足2an+1=an+an+2(n∈正整数),它的前n项和为sn,且a3=-6,s6=-30,.求数列{an}的前n项和的最小值](/uploads/image/z/1071643-67-3.jpg?t=%E5%B7%B2%E7%9F%A5%E6%95%B0%E5%88%97%7Ban%7D%E6%BB%A1%E8%B6%B32an%2B1%3Dan%2Ban%2B2%28n%E2%88%88%E6%AD%A3%E6%95%B4%E6%95%B0%29%2C%E5%AE%83%E7%9A%84%E5%89%8Dn%E9%A1%B9%E5%92%8C%E4%B8%BAsn%2C%E4%B8%94a3%3D-6%2Cs6%3D-30%2C.%E6%B1%82%E6%95%B0%E5%88%97%7Ban%7D%E7%9A%84%E5%89%8Dn%E9%A1%B9%E5%92%8C%E7%9A%84%E6%9C%80%E5%B0%8F%E5%80%BC)
已知数列{an}满足2an+1=an+an+2(n∈正整数),它的前n项和为sn,且a3=-6,s6=-30,.求数列{an}的前n项和的最小值
已知数列{an}满足2an+1=an+an+2(n∈正整数),它的前n项和为sn,且a3=-6,s6=-30,.求数列{an}的前n项和的最小值
已知数列{an}满足2an+1=an+an+2(n∈正整数),它的前n项和为sn,且a3=-6,s6=-30,.求数列{an}的前n项和的最小值
由数列{an}满足2an+1=an+an+2(n∈正整数),
(2an+1=an+an+2看了半天才看明白,最好用下面的方法表示下标)
即a_{n+1}-a_{n}=a_{n+2}-a_{n+1}
知数列为等差数列,设a1=a,增量为b,a3=a+2b,S6=6*(a1+a6)/2=3(2a+5d),解方程组
a+2b=-6
3(2a+5d)=-30
得a=-10,b=2,即数列为
-10,-8,-6,-4,-2,0,2,4,6,……
当n=5或者n=6时都可得到前n项和的最小值-30
由2a(n+1)=a(n)+a(n+2)可得:数列{an}为等差数列
则 S6=6a3+3d=-36+3d=-30
所以 d=2 a1=a3-2d=-10
Sn=n²-11n=(n-11/2)²-121/4
使数列{an}的前n项和的最小值,则n=5或6
所以Sn最小为S5=S6=-30
∵2an+1=an+an+2
∴an+1-an=an+2-an+1
即{an}为等差数列.
∵a3=-6,s6=-30,
∴3(a3+a4)=s6即得a4=-4,
∴d=2,a1=-10,
∴sn=-10n+n(n-1)÷2×2=(n-11/2)^2-121/4
∴sn的最小值为当n=5或6时取得,此时sn=-30.