设数列{an}的前n项和为sn,满足2sn=a(n+1)-2^(n+1)+1,n属于n*.且a1,a2+5,a3成等差数列.1 求a1 （答案1）2.bn=an+2^n 求证bn是等比数列 3.求满足an＞4/5 X 3^的最小正整数n 第三小题求解 an=3^n-2^nan＞4/5 X 3^n
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设数列{an}的前n项和为sn,满足2sn=a(n+1)-2^(n+1)+1,n属于n*.且a1,a2+5,a3成等差数列.1 求a1 （答案1）2.bn=an+2^n 求证bn是等比数列 3.求满足an＞4/5 X 3^的最小正整数n 第三小题求解 an=3^n-2^nan＞4/5 X 3^n
设数列{an}的前n项和为sn,满足2sn=a(n+1)-2^(n+1)+1,n属于n*.且a1,a2+5,a3成等差数列.
1 求a1 （答案1）2.bn=an+2^n 求证bn是等比数列 3.求满足an＞4/5 X 3^的最小正整数n
第三小题求解 an=3^n-2^n
an＞4/5 X 3^n
设数列{an}的前n项和为sn,满足2sn=a(n+1)-2^(n+1)+1,n属于n*.且a1,a2+5,a3成等差数列.1 求a1 （答案1）2.bn=an+2^n 求证bn是等比数列 3.求满足an＞4/5 X 3^的最小正整数n 第三小题求解 an=3^n-2^nan＞4/5 X 3^n
1.
2Sn=a(n+1)-2^(n+1) +1
2S1=2a1=a2 -2²+1=a2-3
a2=2a1 +3
2S2=2a1+2a2=2a1+2(2a1+3)=6a1+6=a3-2³+1=a3-7
a3=6a1+13
a1、a2+5、a3成等差数列,则
2(a2+5)=a1+a3
2(2a1+3+5)=a1+6a1+13
3a1=3
a1=1
2.
2Sn=a(n+1)-2^(n+1) +1=S(n+1)-Sn -2^(n+1) +1
S(n+1)=3Sn +2^(n+1) -1
S(n+1)+2^(n+2) -1/2=3Sn +3×2^(n+1) -3/2=3[Sn +2^(n+1) -1/2]
[S(n+1)+2^(n+2) -1/2]/[Sn +2^(n+1) -1/2]=3,为定值.
S1+2²- 1/2=a1+2² -1/2=1+4 -1/2=9/2,数列{Sn +2^(n+1) -1/2}是以9/2为首项,3为公比的等比数列.
Sn+2^(n+1)-1/2=(9/2)×3^(n-1)=(1/2)×3^(n+1)
Sn=[3^(n+1) +1]/2 -2^(n+1)
n≥2时,
an=Sn-S(n-1)=[3^(n+1) +1]/2 -2^(n+1) -(3ⁿ +1)/2 +2ⁿ=3ⁿ-2ⁿ
n=1时,a1=3-2=1,同样满足通项公式
数列{an}的通项公式为an=3ⁿ-2ⁿ
bn=an+2ⁿ=3ⁿ-2ⁿ+2ⁿ=3ⁿ
b1=a1+2=1+2=3 b(n+1)/bn=3^(n+1)/3ⁿ=3,为定值.
数列{bn}是以3为首项,3为公比的等比数列.
3.
an>(4/5)×3ⁿ
3ⁿ-2ⁿ>(4/5)×3ⁿ
(3/2)ⁿ>5
(3/2)²=2.25>√5 (3/2)⁴>5
(3/2)³=27/8=3.375