设数列{an}的前n项和为Sn,满足2Sn=an+1-2^n+1+1,且a1,a2+5.a3成等差数列(1)求a1的值(2)求数列{an}的通项公式(3)证明:对一切正整数n,有1/a1+1/a2+...1/an
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![设数列{an}的前n项和为Sn,满足2Sn=an+1-2^n+1+1,且a1,a2+5.a3成等差数列(1)求a1的值(2)求数列{an}的通项公式(3)证明:对一切正整数n,有1/a1+1/a2+...1/an](/uploads/image/z/1846797-69-7.jpg?t=%E8%AE%BE%E6%95%B0%E5%88%97%7Ban%7D%E7%9A%84%E5%89%8Dn%E9%A1%B9%E5%92%8C%E4%B8%BASn%2C%E6%BB%A1%E8%B6%B32Sn%3Dan%2B1-2%5En%2B1%2B1%2C%E4%B8%94a1%2Ca2%2B5.a3%E6%88%90%E7%AD%89%E5%B7%AE%E6%95%B0%E5%88%97%EF%BC%881%EF%BC%89%E6%B1%82a1%E7%9A%84%E5%80%BC%EF%BC%882%EF%BC%89%E6%B1%82%E6%95%B0%E5%88%97%7Ban%7D%E7%9A%84%E9%80%9A%E9%A1%B9%E5%85%AC%E5%BC%8F%EF%BC%883%EF%BC%89%E8%AF%81%E6%98%8E%EF%BC%9A%E5%AF%B9%E4%B8%80%E5%88%87%E6%AD%A3%E6%95%B4%E6%95%B0n%2C%E6%9C%891%2Fa1%2B1%2Fa2%2B...1%2Fan)
设数列{an}的前n项和为Sn,满足2Sn=an+1-2^n+1+1,且a1,a2+5.a3成等差数列(1)求a1的值(2)求数列{an}的通项公式(3)证明:对一切正整数n,有1/a1+1/a2+...1/an
设数列{an}的前n项和为Sn,满足2Sn=an+1-2^n+1+1,且a1,a2+5.a3成等差数列
(1)求a1的值
(2)求数列{an}的通项公式
(3)证明:对一切正整数n,有1/a1+1/a2+...1/an
设数列{an}的前n项和为Sn,满足2Sn=an+1-2^n+1+1,且a1,a2+5.a3成等差数列(1)求a1的值(2)求数列{an}的通项公式(3)证明:对一切正整数n,有1/a1+1/a2+...1/an
a1,a2+5,a3成等差数列
a1+a3 = 2(a2+5) (1)
2Sn=a(n+1)-2^(n+1)+1
n=1
2a1 = a2- 4+1
a2= 2a1+3 (2)
n=2
2(a1+a2)=a3-8+1
a3= 2(a1+a2) +7
= 2(a1 +2a1+3) +7
= 6a1+13 (3)
sub (3) ,(2) into (1)
a1 +(6a1+13) = 2(2a1+3+5)
3a1= 3
a1 =1
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{2^(n+1)} - 1/2= 3(Sn + 2{2^n} -1/2)
[S(n+1) +2{2^(n+1)} - 1/2]/(Sn + 2{2^n} -1/2) = 3
(Sn + 2{2^n} -1/2)/(S1 + 2{2^1} -1/2) = 3^(n-1)
Sn + 2{2^n} -1/2 = (3/2).3^n
Sn = 1/2 - 2^(n+1) + (3/2).3^n
an = Sn-S(n-1)
= -2^n + 3^n