等差数列中几个重要结论的证明
09-28
等差数列几个重要结论的证明
(1)等差数列{a n }中,若a n =m ,a m =n (m ≠n ),则a m +n =0 由a n =a m +(n -m )d 得,d =a n -a m m -n ==-1,n -m n -m
∴a m +n =a m +(m +n -m )d =n -n =0
(2)等差数列{a n }中,若S n =m ,S m =n (m ≠n ),则S m +n =-(m +n ) n (n -1) ⎧S =na +d =m 1⎪(n -m )(n +m -1) d , ⎪n 2由⎨得,m -n =(n -m )a 1+2⎪S =ma +m(m-1) d =n m 1⎪⎩2
∴a 1+(n +m -1) d =-1, 2
∴S m +n =(m +n )a 1+(m +n )(m +n -1) d 2
(n +m -1) ⎤⎡=(m +n )⎢a 1+d ⎥ 2⎣⎦
=-(m +n )
(3)等差数列{a n }中,若S n =S m (m ≠n ),则S m +n =0
由S n =S m 知,na 1+n (n -1) m(m-1) d =ma 1+d ,整理得, 22
(n -m )a 1m -n )(m+n -1) (=d ,∴a 21=-(m+n -1) d , 2
∴S m +n =(m +n )a 1+(m +n )(m +n -1) d 2
m +n )(m +n -1) (-(m+n -1) =(m +n )⋅d +d 22
=0