Question

若在直角坐标平面第一象限内，点的坐标（x,y）满足x+y＞n，并且x,y都小于n(n∈N^*)的整点（横、纵坐标均为整数）的个数记为a_n.

(1)求a₃,a₄,a₅并写出数列{a_n}的通项公式a_n=f(n)(不要求证明)；

(2)设列数{b_n}满足b_n=n²-2a_n,T_n=2^n-1b₁+2^n-2b₂+…+2b_n-1+b_n,求T_n.

Answer 1

解：（1）满足条件的点（x,y）在x＞0,表示的平面区域内，如图阴影部分.

?

∴不难得出a₃=1,a₄=2,a₅=3,                                                                              ?

a_n=1+2+3+…+(n-2)?

=.                                                                                                      ?

(2)b_n=n²-2a_n=3n-2,                                                                                                  ?

∵b_n+1-b_n=3.∴数列{b_n}是以1为首项，公差为3的等差数列.                                   ?

T_n=2^n-1b₁+2^n-2b₂+…+2b_n-1+b_n?

=2^n-1+2^n-2·4+…+2(3n-5)+(3n-2),                                                                        ①?

∴2T_n=2ⁿ+2^n-1·4+…+2²（3n-5）+2(3n-2).                                                          ②?

②-①得T_n=2ⁿ+3(2^n-1+2^n-2+…+2)-(3n-2)?

=2ⁿ+-3n+2=2ⁿ⁺²-3n-4.?

∴T_n=2ⁿ⁺²-3n-4.