一个等差数列的前p项和为q,前q项和为p,求其前p+q项之和

Question

一个等差数列的前p项和为q,前q项和为p,求其前p+q项之和

Answer 1

a(n)=a+(n-1)d,
s(n)=na+n(n-1)d/2,
q=s(p)=pa+p(p-1)d/2,q/p=a+(p-1)d/2.
p=s(q)=qa+q(q-1)d/2,p/q=a+(q-1)d/2.
p不等于q.
q/p-p/q=(p-q)d/2.
d/2=[q/p-p/q]/(p-q)=[q^2-p^2]/[pq(p-q)]=-(p+q)/(pq)=-1/q-1/p.
a=q/p-(p-1)d/2 = q/p - (p-1)[-1/p-1/q]=q/p+p/p-1/p +p/q-1/q=q/p+p/q+1-1/p-1/q=[p^2+q^2+pq-p-q]/(pq).
s(p+q)=(p+q)a+(p+q)(p+q-1)d/2
=(p+q)[p^2+q^2+pq-p-q]/(pq) + (p+q)(p+q-1)[-(p+q)/(pq)]
=[(p+q)/(pq)][p^2+q^2+pq-p-q - (p+q)^2 + p+q]
=[(p+q)/(pq)][-pq]
=-p-q