已知数列{an}满足:a1=2,an+1=3an+3的n+1次方-2的n次方(n∈N+)
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已知数列{an}满足:a1=2,an+1=3an+3的n+1次方-2的n次方(n∈N+)
设Cn=an+1/an(n∈N+),是否存在k∈N+,使得Cn≤Ck对一切正整数n均成立,并说明理由
设Cn=an+1/an(n∈N+),是否存在k∈N+,使得Cn≤Ck对一切正整数n均成立,并说明理由
a(n+1)=3an+3^(n+1)-2^n
a(n+1)/3^(n+1) - an/3^n = -(1/3)(2/3)^n
an/3^n - a(n-1)/3^(n-1) = -(1/3)(2/3)^(n-1)
an/3^n - a1/3 = -[ 1-(2/3)^n]
an/3^n = -1/3+ (2/3)^n
an = 2^n - 3^(n-1)
cn = an + 1/an
let
f(x) = 2^x -3^(x-1) + 1/[2^x -3^(x-1)]
f'(x)=(ln2).2^x - (ln3).3^(x-1) - [ (ln2).2^x - (ln3).3^(x-1) ]/1/[2^x -3^(x-1)]^2
f'(x) =1 )
f(1) = 2
f(2) =2
f(3) = -2
{cn} 是递减数列
ie
k =1 or 2
cn
a(n+1)/3^(n+1) - an/3^n = -(1/3)(2/3)^n
an/3^n - a(n-1)/3^(n-1) = -(1/3)(2/3)^(n-1)
an/3^n - a1/3 = -[ 1-(2/3)^n]
an/3^n = -1/3+ (2/3)^n
an = 2^n - 3^(n-1)
cn = an + 1/an
let
f(x) = 2^x -3^(x-1) + 1/[2^x -3^(x-1)]
f'(x)=(ln2).2^x - (ln3).3^(x-1) - [ (ln2).2^x - (ln3).3^(x-1) ]/1/[2^x -3^(x-1)]^2
f'(x) =1 )
f(1) = 2
f(2) =2
f(3) = -2
{cn} 是递减数列
ie
k =1 or 2
cn
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