已知数列{an}满足a1=33,a(n+1)-an=2n,则an/n的最小值为多少

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已知数列{an}满足a1=33,a(n+1)-an=2n,则an/n的最小值为多少
已知数列{an}满足a1=33,a(n+1)-an=2n,则an/n的最小值为多少

已知数列{an}满足a1=33,a(n+1)-an=2n,则an/n的最小值为多少
题目：已知数列{an}满足a1=33,a(n+1)-an=2n,则an/n的最小值
a(n+1)-an=2n
an-a(n-1)=2(n-1)
a(n-1)-a(n-2)=2(n-2)
……（以此类推）
各式相加得,an-a1=2×[1+...+(n-2)+(n-1)]=2×{[1+(n-1)](n-1)}/2=n(n-1)
∴an=a1+n(n-1)=n²-n+33
an/n=(n²-n+33)/n=n+33/n-1
∵a+b≥2√ab（取等号的条件是a=b）
∴n+33/n-1≥2√33-1
n=33/n
∴n²=33
∴n=5或者n=6
a5/5=5+33/5-1=10.6,a6/6=6+33/6-1=10.5

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