数学题、数列题已知:数列{An}的前n项和为Sn,a1=3且当n≥2,n属于N+满足Sn-1是An与-3的等差中项.(1)求a2,a3,a4(2)求数列{An}的通项公式
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![数学题、数列题已知:数列{An}的前n项和为Sn,a1=3且当n≥2,n属于N+满足Sn-1是An与-3的等差中项.(1)求a2,a3,a4(2)求数列{An}的通项公式](/uploads/image/z/693378-18-8.jpg?t=%E6%95%B0%E5%AD%A6%E9%A2%98%E3%80%81%E6%95%B0%E5%88%97%E9%A2%98%E5%B7%B2%E7%9F%A5%EF%BC%9A%E6%95%B0%E5%88%97%7BAn%7D%E7%9A%84%E5%89%8Dn%E9%A1%B9%E5%92%8C%E4%B8%BASn%2Ca1%3D3%E4%B8%94%E5%BD%93n%E2%89%A52%2Cn%E5%B1%9E%E4%BA%8EN%2B%E6%BB%A1%E8%B6%B3Sn-1%E6%98%AFAn%E4%B8%8E-3%E7%9A%84%E7%AD%89%E5%B7%AE%E4%B8%AD%E9%A1%B9.%EF%BC%881%EF%BC%89%E6%B1%82a2%2Ca3%2Ca4%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)
数学题、数列题已知:数列{An}的前n项和为Sn,a1=3且当n≥2,n属于N+满足Sn-1是An与-3的等差中项.(1)求a2,a3,a4(2)求数列{An}的通项公式
数学题、数列题
已知:数列{An}的前n项和为Sn,a1=3且当n≥2,n属于N+满足Sn-1是An与-3的等差中项.(1)求a2,a3,a4(2)求数列{An}的通项公式
数学题、数列题已知:数列{An}的前n项和为Sn,a1=3且当n≥2,n属于N+满足Sn-1是An与-3的等差中项.(1)求a2,a3,a4(2)求数列{An}的通项公式
数列{An}的前n项和为Sn,a1=3且当n≥2,n属于N+满足Sn-1是An与-3的等差中项.(1)求a2,a3,a4(2)求数列{An}的通项公式.
(1)
S1=(A2-3)/2 得A2=3+2*S1=3*2+3=9
S2=(A3-3)/2 得A3=3+2*(A1+A2)=3+2*(3+9)=27
S3=(A4-3)/2 得A4=3+2*(A1+A2+A3)=3+2*(3+9+27)=81
(2)
Sn-1=(An-3)/2
Sn=(A[n+1]-3)/2
两式相减得
An=(A[n+1]-An)/2
3An=A[n+1]
A[n+1]/An=3
它为等比数列
公比为3
An=3^n
是说S的n-1项还是S的n项-1,请补充。
2Sn-1=An-3 得出 2Sn=3An-3 得出a2 = 9,a3 = 27, a4 = 81
猜想 An = 3^n
用书归证明:假设当 n = k 时 2Sn=3An-3 成立
当 n = k+1 时 2Sn+1 = 3An-3 + 2*An+1
= 3^(n+1)-3 + 2*3^(n+1) = 3An+1 - 3
即当 n = k+1 时,假设也成立,故假设成立
即 An = 3^n