Question

已知數(shù)列{a_n}的首項(xiàng)為a₁=1,其前n項(xiàng)和為S_n,且對(duì)任意正整數(shù)n有n,a_n,S_n成等差數(shù)列.
(1)求證:數(shù)列{S_n+n+2}成等比數(shù)列.
(2)求數(shù)列{a_n}的通項(xiàng)公式.

Answer 1

(1)見(jiàn)解析   (2) a_n=2ⁿ-1

(1)因?yàn)閚,a_n,S_n成等差數(shù)列,所以2a_n=S_n+n,由當(dāng)n≥2時(shí),a_n=S_n-S_n-1,
所以2(S_n-S_n-1)=S_n+n,
即S_n=2S_n-1+n(n≥2),
所以S_n+n+2=2S_n-1+2n+2=2[S_n-1+(n-1)+2].
又S₁+1+2=4≠0,
所以=2,所以數(shù)列{S_n+n+2}成等比數(shù)列.
(2)由(1)知{S_n+n+2}是以S₁+3=a₁+3=4為首項(xiàng),2為公比的等比數(shù)列,所以S_n+n+2=4×2^n-1=2ⁿ⁺¹,又2a_n=n+S_n,所以2a_n+2=2ⁿ⁺¹,所以a_n=2ⁿ-1.