已知数列{a n },a 1 =1,a n =3 n-1 a n-1 (n≥2,n∈N*).(1)求数列{a n }的通项公式;(2)设 S

2025-05-20 14:36:06
推荐回答(1个)
回答1:

(1)由已知得,当n≥2时,
a n
a n-1
= 3 n-1

a n =
a n
a n-1
?
a n-1
a n-2
?…?
a 3
a 2
?
a 2
a 1
? a 1

= 3 n-1 ? 3 n-2 ?…? 3 2 ? 3 1 ?1= 3 (n-1)+(n-2)+…+1 = 3
n(n-1)
2

(2) S n =lo g 3 (
a n
27 3n
)

= lo g 3
3
n(n-1)
2
27 3n
=
n(n-1)
2
-9n=
n 2 -19n
2

b 1 =S 1 =-9;
当n≥2时,b n =f(n)-f(n-1)=n-10,
上式中,当n=1时,n-10=-9=b 1
∴b n =n-10.
(3)数列{b n }为首项为-9,公差为1的等差数列,且当n≤10时,b n ≤0,故n≤10时,T n =|S n |=
19n- n 2
2

当n>10时,T n =|b 1 |+|b 2 |+|b 3 |+…+|b n |
=-b 1 -b 2 -…-b 10 +b 11 +…+b n
=|b 1 +b 2 +b 3 +b 4 +…+b n |+2|b 1 +b 2 +…+b 10 |
=
n 2 -19n+180
2

∴T n =
19n- n 2
2
,(n≤10,n∈N*)
n 2 -19n+180
2
,(n>10,n∈N*).