等差数列{an}中,前p项的和与前q项的和相等,p不等于q,求前p+q项的和S(p+q)

Question

等差数列{an}中,前p项的和与前q项的和相等,p不等于q,求前p+q项的和S(p+q)

Answer 1

Sn=na1+n(n-1)d/2=na1+d/2*(n²-n)
那么Sp=pa1+d/2*(p²-p)
Sq=qa1+d/2*(q²-q)
所以Sp-Sq=a1*(p-q)+d/2*(p²-p-q²+q)
=a1*(p-q)+d/2*[(p+q)(p-q)-(p-q)]
=a1*(p-q)+d/2*(p-q)(q+q-1)
=(p-q)*[a1+d/2*(p+q-1)]
=0
而p≠q,即p-q≠0,所以a1+d/2*(p+q-1)=0
S(p+q)=(p+q)*a1+d/2*(p+q)(p+q-1)
=(p+q)*[a1+d/2*(p+q-1)]
=(p+q)*0
=0