Question

已知數(shù)列{a_n}的前n項(xiàng)和S_n滿足a_n+2S_nS_n-1="0" (n≥2),a₁=,求a_n.

Answer 1

a_n=.

∵當(dāng)n≥2時(shí)，a_n=S_n-S_n-1,
∴S_n-S_n-1+2S_nS_n-1=0，
即-=2，                                                       4分
∴數(shù)列是公差為2的等差數(shù)列.                                      6分
又S₁=a₁=，∴=2，
∴=2+（n-1）·2=2n，
∴S_n=.                                                               10分
∴當(dāng)n≥2時(shí)，a_n=-2S_nS_n-1=-2··
=-，                                                            12分
∴a_n=.                                                14分