已知数列{an}满足a1=3,a(n+1)=an+3n方+3n+2-1/n(n+1),n是正整数
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(1)
a(n+1)=an+3n²+3n+2 -1/[n(n+1)]=an+(n+1)³-n³-1/n +1/(n+1) +1
[a(n+1) -(n+1)³ -1/(n+1)]-(an-n³ -1/n)=1,为定值
a1 -1³ -1/1=3-1-1=1
数列{an -n³ -1/n}是以1为首项,1为公差的等差数列
an -n³ -1/n=1+1×(n-1)=n
an=n³+n +1/n
n=1时,a1=1+1 +1/1=3,同样满足通项公式
数列{an}的通项公式为an=n³+n +1/n
(2)
an=n³+n +1/n=(n⁴+n²+1)/n
n>0 an>0
[1/a(n+1)]/(1/an)=[(n+1)(n⁴+n²+1)]/[(n+1)⁴+(n+1)²+1]n
太繁琐了,不写了,化简结果进行缩放:
[1/a(n+1)]/(1/an)<1/3
1/a1+1/a2+...+1/an
<1/3+1/3²+...+1/3ⁿ
=(1/3)(1-1/3ⁿ)/(1-1/3)
=(1/2)(1-1/3ⁿ
=1/2 -1/(2×3ⁿ)
<1/2-0
=1/2
a(n+1)=an+3n²+3n+2 -1/[n(n+1)]=an+(n+1)³-n³-1/n +1/(n+1) +1
[a(n+1) -(n+1)³ -1/(n+1)]-(an-n³ -1/n)=1,为定值
a1 -1³ -1/1=3-1-1=1
数列{an -n³ -1/n}是以1为首项,1为公差的等差数列
an -n³ -1/n=1+1×(n-1)=n
an=n³+n +1/n
n=1时,a1=1+1 +1/1=3,同样满足通项公式
数列{an}的通项公式为an=n³+n +1/n
(2)
an=n³+n +1/n=(n⁴+n²+1)/n
n>0 an>0
[1/a(n+1)]/(1/an)=[(n+1)(n⁴+n²+1)]/[(n+1)⁴+(n+1)²+1]n
太繁琐了,不写了,化简结果进行缩放:
[1/a(n+1)]/(1/an)<1/3
1/a1+1/a2+...+1/an
<1/3+1/3²+...+1/3ⁿ
=(1/3)(1-1/3ⁿ)/(1-1/3)
=(1/2)(1-1/3ⁿ
=1/2 -1/(2×3ⁿ)
<1/2-0
=1/2
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