奇偶分开算（2020课标Ⅰ（文），16）

2023-04-18 11:20 作者:数学老顽童 0人读过 | 我要投稿

（2020课标Ⅰ（文），16）数列 $%5Cleft%5C%7B%20a_n%20%5Cright%5C%7D%20$ 满足 $a_%7Bn%2B2%7D%2B%5Cleft(%20-1%20%5Cright)%20%5Ena_n%3D3n-1$ ，前 $16$ 项和为 $540$ ，则 $a_1%3D$ ________.

解：当 $n$ 为偶数时，

$a_%7Bn%2B2%7D%2Ba_n%3D3n-1$ ，所以

$%5Cbegin%7Baligned%7D%09%26a_4%2Ba_2%2Ba_8%2Ba_6%2Ba_%7B12%7D%2Ba_%7B10%7D%2Ba_%7B16%7D%2Ba_%7B14%7D%5C%5C%09%3D%263%5Ctimes%202-1%2B3%5Ctimes%206-1%2B3%5Ctimes%2010-1%2B3%5Ctimes%2014-1%5C%5C%09%3D%26%5Ccolor%7Bred%7D%7B92%7D%5C%5C%5Cend%7Baligned%7D$

当 $n$ 为奇数时，

$a_%7Bn%2B2%7D-a_n%3D3n-1$ ，

$a_n-a_%7Bn-2%7D%3D3%5Cleft(%20n-2%20%5Cright)%20-1$ ，

$a_%7Bn-2%7D-a_%7Bn-4%7D%3D3%5Cleft(%20n-4%20%5Cright)%20-1$ ，

……

$a_7-a_5%3D3%5Ctimes%205-1$ ，

$a_5-a_3%3D3%5Ctimes%203-1$ ，

$a_3-a_1%3D3%5Ctimes%201-1$ ，

累加可得

$%5Cbegin%7Baligned%7D%0A%09a_%7Bn%2B2%7D-a_1%26%3D%5Cfrac%7B%5Cleft(%202%2B3n-1%20%5Cright)%20%5Ccdot%20%5Cfrac%7Bn%2B1%7D%7B2%7D%7D%7B2%7D%5C%5C%0A%09%26%3D%5Cfrac%7B3%7D%7B4%7Dn%5E2%2Bn%2B%5Cfrac%7B1%7D%7B4%7D%5C%5C%0A%5Cend%7Baligned%7D$

所以 $%5Ccolor%7Bred%7D%7Ba_%7Bn%2B2%7D%3Da_1%2B%5Cfrac%7B3%7D%7B4%7Dn%5E2%2Bn%2B%5Cfrac%7B1%7D%7B4%7D%7D$ .

所以

$%5Cbegin%7Baligned%7D%09%26a_1%2Ba_3%2Ba_5%2B%5Ccdots%20%2Ba_%7B15%7D%5C%5C%09%3D%268a_1%2B%5Cfrac%7B3%7D%7B4%7D%5Ctimes%20%5Cleft(%201%5E2%2B3%5E2%2B5%5E2%2B%5Ccdots%20%2B13%5E2%20%5Cright)%5C%5C%09%2B%26%5Cleft(%201%2B3%2B5%2B%5Ccdots%20%2B13%20%5Cright)%20%2B%5Cfrac%7B1%7D%7B4%7D%5Ctimes%207%5C%5C%09%3D%26%5Ccolor%7Bred%7D%7B8a_1%2B392%7D%5C%5C%5Cend%7Baligned%7D$