Question

已知數(shù)列{a_n}、{b_n}滿足：b_n=,求證：數(shù)列{a_n}成等差數(shù)列的充要條件是數(shù)列{b_n}也是等差數(shù)列.

Answer 1

①必要性：

設(shè){a_n}成等差數(shù)列，公差為d,∵{a_n}成等差數(shù)列.

     從而b_n+1－b_n=a₁+n·d－a₁－(n－1) d=d為常數(shù).

    故{b_n}是等差數(shù)列，公差為d.

②充分性:

設(shè){b_n}是等差數(shù)列，公差為d′,則b_n=(n－1)d′

∵b_n(1+2+…+n)=a₁+2a₂+…+na_n                                                                                ①

b_n－1(1+2+…+n－1)=a₁+2a₂+…+(n－1)a_n                                                                   ②

①－②得：na_n=b_n－1?

∴a_n=,從而得a_n+1－a_n=d′為常數(shù)，故{a_n}是等差數(shù)列.

綜上所述，數(shù)列{a_n}成等差數(shù)列的充要條件是數(shù)列{b_n}也是等差數(shù)列.

解析:

同答案