已知数列｛an｝中，a1=1，a1+2a2+3a3+…+nan=(n+1)/2an+1（n∈N*） 在数列{an}中,a1=1,a1+2a2+3a3+....+...
\u5df2\u77e5\u6570\u5217{an}\u4e2d\uff0ca1=1,a1+2a2+3a3+\u2026\u2026+nan=(n+1)/2*an+1 \u6c42\u6570\u5217{an}\u7684\u901a\u9879an\u9996\u5148\u9ebb\u70e6\u697c\u4e3b\u4e0b\u6b21\u628a\u4e0b\u6807\u6807\u597d\uff0c\u81f3\u5c11\u6253\u4e2a\u62ec\u53f7\uff0c\u5982a(n+1),\u7ecf\u8fc7\u9a8c\u8bc1\uff0c\u662f\u8868\u793a\u8fd9\u79cd\u610f\u601d\uff0c\u5426\u5219\u540e\u9762\u7684\u9879\u90fd\u662f0
\u5bf9\u4e8e\u8fd9\u9053\u9898\u80af\u5b9a\u6709\u8bb8\u591a\u4eba\u6ca1\u770b\u6e05\u695a\u9898\u76ee\uff0c\u5176\u5b9e\u662f\u8fd9\u6837\u7684\uff1a
\u8fd9\u9898\u7b54\u6848\u662f a(1)=a(2)=1,a(n)=2/n*3^(n-2)
\u6211\u7b80\u8981\u5730\u8bf4\u4e00\u4e0b
\u5bf9\u4e8e\u9898\u76ee\u7684\u7b49\u5f0f\uff0c\u53d8\u91cf\u5206\u522b\u53d6n\u548cn-1\u5f97\u4e24\u4e2a\u5f0f\u5b50\uff0c\u76f8\u51cf\u5316\u7b80\u5f97\u5230a(n)/a(n-1)=(n-2)/(n-1).
\u6ce8\u610f\u5230a(1)=a(2)=1,a(3)=2,a(3)/a(2)\u662f\u6ee1\u8db3\u6b64\u6761\u4ef6\u7684\u8d77\u59cb\u9879\uff0c\u7136\u540e\u7d2f\u4e58\u5c31\u5f97\u5230\u60f3\u8981\u7684\u7b54\u6848\uff0c\u6ce8\u610f\u9879\u6570
a1=1,a1+2a2+3a3+....+nan=(n+1)(a(n+1))/2,
\u4ee4n=1\u5f97\uff1aa1=2a2/2, a2=1.
\u5f53n\u22652\u65f6\uff0ca1+2a2+3a3+....+(n-1)a(n-1)=na(n)/2,
\u4e24\u5f0f\u76f8\u51cf\u5f97\uff1anan=(n+1)(a(n+1))/2 -na(n)/2,
3na(n)/2=(n+1)(a(n+1))/2\uff0c
a(n+1) /a(n)= 3n/(n+1)( n\u22652),
\u6240\u4ee5a3/a2=3•2/3,
a4/a3=3•3/4,
a5/a4=3•4/5,
\u2026\u2026\u2026\u2026
a(n) /a(n-1)= 3(n-1)/n
\u4ee5\u4e0a\u5404\u5f0f\u76f8\u4e58\u5f97\uff1aa(n) / a2=3^(n-2)•2/n( n\u22652),
a(n)=3^(n-2)•2/n ( n\u22652),
\u7efc\u4e0a\u53ef\u77e5\uff1an=1\u65f6\uff0ca(n)=1.
n\u22652\u65f6\uff0ca(n)=3^(n-2)•2/n.
n≥2时，
a1+2a2+3a3+...+nan=[(n+1)/2]a(n+1) (1)
a1+2a2+3a3+...+(n-1)a(n-1)=(n/2)an (2)
(1)-(2)
nan=[(n+1)/2]a(n+1)-(n/2)an
(n+1)a(n+1)=3nan
[(n+1)a(n+1)]/(nan)=3，为定值
a1×1=1×1=1，数列{nan}是以1为首项，3为公比的等比数列
nan=1×3^(n-1)=3^(n-1)
an=3^(n-1)/n
n=1时，a1=1/1=1，同样满足通项公式
数列{an}的通项公式为an=3^(n-1)/n
2.
n^2·an=n^2·[3^(n-1)/n]=n·3^(n-1)
Tn=1×1+2×3+3×3²+...+n×3^(n-1)
3Tn=1×3+2×3²+...+(n-1)×3^(n-1)+n×3ⁿ
Tn-3Tn=-2Tn=1+3+...+3^(n-1)-n×3ⁿ
=1×(3ⁿ-1)/(3-1) -n×3ⁿ
=[(1-2n)×3ⁿ-1]/2
Tn=[(2n-1)×3ⁿ +1]/4
3.
[a(n+1)/(n+2)]/[an/(n+1)]
=[3ⁿ/(n+1)(n+2)]/[3^(n-1)/n(n+1)]
=3n/(n+2)
n≥1 n/(n+2)≤1/3，当且仅当n=1时取等号
3n/(n+2)≤1，当且仅当n=1时取等号
即对数列{an/(n+1)}， a1/2=a2/3，当n≥2时，{an/(n+1)}单调递减
a1/2=1/2
要不等式an≤(n+1)λ对任意正整数n恒成立，即an/(n+1)≤λ对任意正整数n恒成立，只需当an/(n+1)取最大值时不等式成立。
λ≥1/2，λ的最小值为1/2
a(1)=1,
a(1)+2a(2)+...+na(n) = (n+1)a(n+1)/2, a(1) = 2a(2)/2, a(2)=1.
a(1)+2a(2)+...+na(n)+(n+1)a(n+1) = (n+2)a(n+2)/2 = (n+1)a(n+1)/2 + (n+1)a(n+1)=3(n+1)a(n+1)/2,
(n+2)a(n+2) = 3(n+1)a(n+1),
{(n+1)a(n+1)}是首项为2a(2)=2, 公比为3的等比数列。
(n+1)a(n+1) = 2*3^(n-1) = (2/9)3^(n+1)
a(n+1) = (2/9)3^(n+1)/(n+1),
{a(n)}的通项公式为:
a(1)=1,
n>=2时，a(n) = (2/9)3^n/n = (2*3^n)/(9n)
b(n) = n^2a(n),
b(1) = a(1)=1,
n>=2时，b(n)=(2n/9)*3^n = 2n*3^(n-2),
t(1)=b(1)=1,
n>=2时，
t(n) = b(1)+b(2)+b(3)+...+b(n-1)+b(n)
= 1 + 2*2*1 + 2*3*3 + ... + 2(n-1)*3^(n-3) + 2n*3^(n-2),
3t(n) = 3 + 2*2*3 + 2*3*3^2 + ... + 2(n-1)*3^(n-2) + 2n*3^(n-1),
2t(n) = 3t(n)- t(n) = 2 - 2*2 - 2*3 - ... - 2*3^(n-2) + 2n*3^(n-1)
= 2n*3^(n-1) - 2*1 - 2*3 - ... - 2*3^(n-2)
= 2n*3^(n-1) - 2[1+3+...+3^(n-2)]
= 2n*3^(n-1) - 2[3^(n-1)-1]/(3-1)
= 2n*3^(n-1) - 2[3^(n-1)-1]/2,
t(n) = n*3^(n-1) - [3^(n-1)-1]/2 = [(2n-1)*3^(n-1) + 1]/2.
t(1)=1,
n>=2时，t(n) = [(2n-1)*3^(n-1) + 1]/2.
c(n) = a(n)/(n+1),
c(1) = a(1)/2 = 1/2.
n>=2时，c(n) = (2/9)3^n/[n(n+1)] >0.
c(n+1) = (2/9)3^(n+1)/[(n+1)(n+2)],
c(n+1)/c(n) = 3n/(n+2) = (n+2-2+2n)/(n+2) = 1 + 2(n-1)/(n+2)>1.
n>=2时，c(n+1)>c(n), {c(n),n>=2}单调递增 。c(n)>=c(2)=(2/9)3^2/[2*3] = 1/3.
c(1)=1/2>1/3.
因此，Lamda 的最小值=1/3.
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绛旓細a1=1,a2=2,log₂a(n+1)+log2₂an=n log₂[a(n+1)an]=n a(n+1)an=2^n 鈭碼(n+1)=2^n/an a(n+2)=2^(n+1)/a(n+1)=2^(n+1)/[2^n/an]=2*an 鈭碼(n+2)/an=2 鍗砤1,a3,a5,...,a2013鏋勬垚绛夋瘮鏁板垪锛鍏瘮涓2,鍏1007椤 a2,a4,a6,...锛...
绛旓細璇佹槑a(n+1)+1=2(an+1)鎵浠ャ恆(n+1)+1銆/(an+1)=2 {an+1}鏄互a1+1=2涓洪椤,鍏瘮涓2鐨勭瓑姣鏁板垪 鎵浠(an)+1=2 x2^(n--1)an=2^n--1 Sn=(2+4+8+.+2^n)---n =2^(n+1)-n-1
