5.等比數(shù)列的性質(zhì):
(1)當(dāng)時(shí),則有,特別地,當(dāng)時(shí),則有.
(2) 若是等比數(shù)列,且公比,則數(shù)列 ,…也是等比數(shù)列。
當(dāng),且為偶數(shù)時(shí),數(shù)列
,…是常數(shù)數(shù)列0,它不是等比數(shù)列.
(3)若,則為遞增數(shù)列;
若, 則為遞減數(shù)列;
若 ,則為遞減數(shù)列;
若, 則為遞增數(shù)列;
若,則為擺動(dòng)數(shù)列;
若,則為常數(shù)列.
(4)當(dāng)時(shí),,這里,但,這是等比數(shù)列前項(xiàng)和公式特征,據(jù)此判斷數(shù)列是否為等比數(shù)列。
(5) 在等比數(shù)列中,當(dāng)項(xiàng)數(shù)為偶數(shù)時(shí),;項(xiàng)數(shù)為奇數(shù)時(shí),.
(7)數(shù)列既成等差數(shù)列又成等比數(shù)列,那么數(shù)列是非零常數(shù)數(shù)列,故常數(shù)數(shù)列僅是此數(shù)列既成等差數(shù)列又成等比數(shù)列的必要非充分條件。