问题
解答题
在等差数列{an}中,a1=8,a3=4. (1)求数列{an}的通项公式; (2)设Sn=|a1|+|a2|+…+|an|,求Sn; (3)设bn=
|
答案
(1)∵{an}成等差数列,a1=8,a3=4.
∴8+2d=4,解得公差d=-2
∴an=8+(n-1)×(-2)=10-2n.
(2)设a1+a2+…+an=S'n
由an=10-2n≥0 得n≤5,
∴当n≤5时,Sn=|a1|+|a2|+…+|an|=a1+a2+…+an=
=-n2+9n=S'n.n(8+10-2n) 2
当n>5时,Sn=|a1|+|a2|+…+|an|=a1+a2+…+a5-a6-…-an
=2S'5-S'n=n2-9n+40.
故Sn=-n2+9n n2-9n+40
(n∈N)1≤n≤5 n>5
(3)bn=
=1 n(12-an)
=1 n•(2n+2)
(1 2
-1 n
)1 n+1
∴Tn=b1+b2+…+bn=
[(1-1 2
)+(1 2
-1 2
)+…+(1 3
-1 n
)]=1 n+1
.n 2(n+1)