求n值,有一数列：A1,A2,A3,A4...An-1,An,规定A1=2,A2-A1=4,A3-A2=6,...,An-An-1=2n.当1/A2+1/A3+1/A4+...+1/An的结果是1005/2012,n的值是?

Question

求n值,
有一数列：A1,A2,A3,A4...An-1,An,规定A1=2,A2-A1=4,A3-A2=6,...,An-An-1=2n.当1/A2+1/A3+1/A4+...+1/An的结果是1005/2012,n的值是?

Answer 1

A(n)-A(n-1)=2n
A(n-1)-A(n-2)=2n-2
…………
A(2)-A(1)=4
A(1)=2
全加起来,得：A(n)=2n+2n-2+……+4+2=n(n+1)
故1/A(n)=1/n(n+1)=1/n-1/(n+1)
1/A(2)+1/A(3)+……+1/A(n)
=1/2-1/3+1/3-1/4+……+1/n-1/(n+1)
=1/2-1/(n+1)
=n/2(n+1)=1005/2012
解得：
n=1005