数列 (27 11:16:31)已知数列{an}中,a1=2,a(n+1)=4an-3n+1(n∈N+),bn=an-n 求数列{an}的前n项和
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![数列 (27 11:16:31)已知数列{an}中,a1=2,a(n+1)=4an-3n+1(n∈N+),bn=an-n 求数列{an}的前n项和](/uploads/image/z/400141-37-1.jpg?t=%E6%95%B0%E5%88%97+%2827+11%3A16%3A31%29%E5%B7%B2%E7%9F%A5%E6%95%B0%E5%88%97%7Ban%7D%E4%B8%AD%2Ca1%3D2%2Ca%28n%2B1%29%3D4an-3n%2B1%28n%26isin%3BN%2B%29%2Cbn%3Dan-n+%E6%B1%82%E6%95%B0%E5%88%97%7Ban%7D%E7%9A%84%E5%89%8Dn%E9%A1%B9%E5%92%8C)
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数列 (27 11:16:31)已知数列{an}中,a1=2,a(n+1)=4an-3n+1(n∈N+),bn=an-n 求数列{an}的前n项和
数列 (27 11:16:31)
已知数列{an}中,a1=2,a(n+1)=4an-3n+1(n∈N+),bn=an-n 求数列{an}的前n项和
数列 (27 11:16:31)已知数列{an}中,a1=2,a(n+1)=4an-3n+1(n∈N+),bn=an-n 求数列{an}的前n项和
因为bn=an-n,a(n+1)=4an-3n+1
所以b(n+1)=a(n+1)-n-1 =4an-3n+1-n-1
=4(an-n)=4bn
因为b1=a1-1=2-1=1
所以bn为首项1,公比4的等比数列
即bn=an-n=4^(n-1)
所以an=4^(n-1)+n
Sn=a1+a2+……+an
=4^0+1+4^1+2+……+4^(n-1)+n
=(4^(n-1)-1)/3+(n^2+n)/2