设等差数列共2n+1项,公差为d,则偶数项为公差2d的等差数列,有n项,奇数项为公差为2d的等差数列,有n+1项,中间项为a(n+1).
S偶=na2+2dn(n-1)/2=na2+dn(n-1)=n(a1+d)+dn(n-1)=na1+dn+dn²-dn=dn²+na1=33 (1)
S奇=(n+1)a1+2dn(n+1)/2=(n+1)a1+dn²+dn=44 (2)
(2)-(1)
a1+nd=11
a(n+1)=11
S=a1+a2+...+a(2n+1)
=[a1+a(2n+1)]+[a2+a(2n)]+...+[an+a(n+2)]+a(n+1)
=2a(n+1)+2a(n+1)+...+2a(n+1)+a(n+1)
=(2n+1)a(n+1)
=11(2n+1)=44+33=77
2n+1=7
n=3
中间项为11,项数为3项.