已知等差数列{an}的公差d≠0,它的前n项和为Sn,若S5=70,且a2,a7,a22成等比数列.(Ⅰ)求数列{an}的

2025-05-15 20:37:06
推荐回答(1个)
回答1:

(Ⅰ)解:依题意,有

S5=70
a72a2a22
,即
5a1+10d=70
(a1+6d)2=(a1+d)(a1+21d)

解得a1=6,d=4,
∴数列{an}的通项公式为an=4n+2(n∈N*).
(Ⅱ)证明:由(Ⅰ)可得Sn=2n2+4n,
an+6
(n+1)Sn
4n+2+6
(n+1)(2n2+4n)
4(n+2)
2n(n+1)(n+2)
2
n(n+1)

Tn=2[(1?
1
2
)+(
1
2
?
1
3
)+…+(
1
n
?
1
n+1
)]=2(1?
1
n+1
)

{
1
n+1
}
是递减数列,且n∈N*
0<
1
n+1
1
2
.∴?