已知数列{an}满足a1=1,a2=20,a(n+2)=2a(n+1)+8an,求an通项和前n项和Sn,补充:第一小题已证明{a(n+1)+2an}是等比数列 且a(n+2)中n+2为下标
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![已知数列{an}满足a1=1,a2=20,a(n+2)=2a(n+1)+8an,求an通项和前n项和Sn,补充:第一小题已证明{a(n+1)+2an}是等比数列 且a(n+2)中n+2为下标](/uploads/image/z/1270331-35-1.jpg?t=%E5%B7%B2%E7%9F%A5%E6%95%B0%E5%88%97%7Ban%7D%E6%BB%A1%E8%B6%B3a1%3D1%2Ca2%3D20%2Ca%28n%2B2%29%3D2a%28n%2B1%29%2B8an%2C%E6%B1%82an%E9%80%9A%E9%A1%B9%E5%92%8C%E5%89%8Dn%E9%A1%B9%E5%92%8CSn%2C%E8%A1%A5%E5%85%85%EF%BC%9A%E7%AC%AC%E4%B8%80%E5%B0%8F%E9%A2%98%E5%B7%B2%E8%AF%81%E6%98%8E%7Ba%28n%2B1%29%2B2an%7D%E6%98%AF%E7%AD%89%E6%AF%94%E6%95%B0%E5%88%97+%E4%B8%94a%28n%2B2%29%E4%B8%ADn%2B2%E4%B8%BA%E4%B8%8B%E6%A0%87)
已知数列{an}满足a1=1,a2=20,a(n+2)=2a(n+1)+8an,求an通项和前n项和Sn,补充:第一小题已证明{a(n+1)+2an}是等比数列 且a(n+2)中n+2为下标
已知数列{an}满足a1=1,a2=20,a(n+2)=2a(n+1)+8an,求an通项和前n项和Sn,补充:第一小题已证明{a(n+1)+2an}是等比数列 且a(n+2)中n+2为下标
已知数列{an}满足a1=1,a2=20,a(n+2)=2a(n+1)+8an,求an通项和前n项和Sn,补充:第一小题已证明{a(n+1)+2an}是等比数列 且a(n+2)中n+2为下标
a(n+2)=2a(n+1)+8a(n),
r^2 = 2r+8,0=r^2 - 2r - 8 = (r-4)(r+2),r=4 或r=-2.
a(n+2) + 2a(n+1) = 4a(n+1) + 8a(n) = 4[a(n+1)+2a(n)],
{a(n+1)+2a(n)}是首项为a(2)+2a(1)=22,公比为4的等比数列.
a(n+1)+2a(n)=22*4^(n-1),.(*)
a(n+2) - 4a(n+1) = -2a(n+1) + 8a(n) = -2[a(n+1) - 4a(n)],
{a(n+1) - 4a(n)}是首项为a(2)-4a(1)=16,公比为(-2)的等比数列.
a(n+1)-4a(n) = 16*(-2)^(n-1),.(**)
式(*) - (**),有,
2a(n) + 4a(n) = 6a(n) = 22*4^(n-1) - 16*(-2)^(n-1),
a(n) = [11*4^(n-1) - 8*(-2)^(n-1)]/3,
s(n) = (11/3)[1+4+...+4^(n-1)] + (-8/3)[1+(-2) + ...+ (-2)^(n-1)]
=(11/3)[4^n - 1]/(4-1) -(8/3)[(-2)^n - 1]/(-2-1)
=(11/9)[4^n - 1] + (8/9) [ (-2)^n - 1]
=(11/9)4^n + (8/9)(-2)^n - 19/9