| an−1 |
| an−1+2 |
anan-1+2an=an-1，
即1+
| 2 |
| an−1 |
| 1 |
| an |
∴
| 1 |
| an |
| 1 |
| an−1 |
∴数列{
| 1 |
| an |
| 1 |
| a1 |
则
| 1 |
| an |
an＝
| 1 |
| 2n−1 |
由ak＞
| 1 |
| 2009 |
| 1 |
| 2k−1 |
| 1 |
| 2009 |
即2k＜2010．
∵k为正整数，
∴k的最大值为10．
故选：D．
| an−1 |
| an−1+2 |
| 1 |
| 2009 |
| an−1 |
| an−1+2 |
| 2 |
| an−1 |
| 1 |
| an |
| 1 |
| an |
| 1 |
| an−1 |
| 1 |
| an |
| 1 |
| a1 |
| 1 |
| an |
| 1 |
| 2n−1 |
| 1 |
| 2009 |
| 1 |
| 2k−1 |
| 1 |
| 2009 |