等差数列an中,若m,n,l,k∈N,且m+n=k+l,则--------反之-------；特别的,如果m+n=2k,则-------

Question

等差数列an中,若m,n,l,k∈N,且m+n=k+l,则--------反之-------；特别的,如果m+n=2k,则-------

Answer 1

m+n=k+l
am+an=a1+(m-1)d+a1+(n-1)d
=2a1-2d+(m+n)d
=2a1-2d+(k+l)d
=a1+(k-1)d+a1+(l-1)d
=ak+al
若am+an=ak+al
a1+(m-1)d+a1+(n-1)d=a1+(k-1)d+a1+(l-1)d
所以m+n=k+l
所以
m+n=k+l,则ak+al
反之若am+an=ak+al则m+n=k+l
特别的,如果m+n=2k,则am++an=2ak