a(1)=0, a(2n+1)=a(2n-1)+2,
{a(2n-1)}是首项为0,公差为2的等差数列.
a(2n-1)=0+2(n-1)=2(n-1),
t(n)=a(1)+a(3)+...+a(2n-1)=n(n-1).
a(2)=2, a[2(n+1)]=a(2n)+2,
{a(2n)}是首项为2,公差为2的等差数列.
a(2n)=2+2(n-1),
w(n)=a(2)+a(4)+...+a(2n)=2n+n(n-1),
s(2n)=a(1)+a(2)+...+a(2n-1)+a(2n)=t(n)+w(n)=2n+2n(n-1)=2n^2,
s(1)=t(1)=0,
s(2n+1)=a(1)+a(2)+...+a[2n]+a(2n+1)=w(n)+t(n+1)=2n+n(n-1)+(n+1)n=2n+2n^2=2n(n+1).
若5000=s(N)=2n^2, 2500= n^2, n=50, N=2n=100.满足要求.
若5000=s(N)=2n(n+1), 2500=n(n+1),n无整数解.
因此,N=100.
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