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函數(shù)也遞推(2022新高考Ⅱ,8)

2023-02-16 10:07 作者:數(shù)學(xué)老頑童  | 我要投稿

(2022新高考Ⅱ,8)已知函數(shù)f%5Cleft(%20x%20%5Cright)%20的定義域?yàn)?img type="latex" class="latex" src="http://api.bilibili.com/x/web-frontend/mathjax/tex?formula=%5Cmathbf%7BR%7D" alt="%5Cmathbf%7BR%7D">,且f%5Cleft(%20x%2By%20%5Cright)%20%2Bf%5Cleft(%20x-y%20%5Cright)%20%3Df%5Cleft(%20x%20%5Cright)%20f%5Cleft(%20y%20%5Cright)%20,f%5Cleft(%201%20%5Cright)%20%3D1,則%5Csum_%7Bk%3D1%7D%5E%7B22%7D%7Bf%5Cleft(%20k%20%5Cright)%7D%3D(? ? )

A.-3

B.-2

C.0

D.1

解:令%5Ccolor%7Bred%7D%7By%3D1%7D,可得

f%5Cleft(%20x%2B1%20%5Cright)%20%2Bf%5Cleft(%20x-1%20%5Cright)%20%3Df%5Cleft(%20x%20%5Cright)%20f%5Cleft(%201%20%5Cright)%20,

f%5Cleft(%20x%2B1%20%5Cright)%20%3Df%5Cleft(%20x%20%5Cright)%20-f%5Cleft(%20x-1%20%5Cright)%20,

%5Ccolor%7Bred%7D%7Bf%5Cleft(%20x%20%5Cright)%20%3Df%5Cleft(%20x-1%20%5Cright)%20-f%5Cleft(%20x-2%20%5Cright)%20%7D;

%5Ccolor%7Bred%7D%7By%3D0%7D,可得

f%5Cleft(%20x%20%5Cright)%20%2Bf%5Cleft(%20x%20%5Cright)%20%3Df%5Cleft(%20x%20%5Cright)%20f%5Cleft(%200%20%5Cright)%20

%5Cleft%5B%20f%5Cleft(%200%20%5Cright)%20-2%20%5Cright%5D%20%5Ccdot%20f%5Cleft(%20x%20%5Cright)%20%3D0,

所以%5Ccolor%7Bred%7D%7Bf%5Cleft(%200%20%5Cright)%20%3D2%7D

所以

%5Ccolor%7Bred%7D%7Bf%5Cleft(%202%20%5Cright)%20%3D%7Df%5Cleft(%201%20%5Cright)%20-f%5Cleft(%200%20%5Cright)%20%3D1-2%3D%5Ccolor%7Bred%7D%7B-1%7D

%5Ccolor%7Bred%7D%7Bf%5Cleft(%203%20%5Cright)%20%3D%7Df%5Cleft(%202%20%5Cright)%20-f%5Cleft(%201%20%5Cright)%20%3D-1-1%3D%5Ccolor%7Bred%7D%7B-2%7D

%5Ccolor%7Bred%7D%7Bf%5Cleft(%204%20%5Cright)%20%3D%7Df%5Cleft(%203%20%5Cright)%20-f%5Cleft(%202%20%5Cright)%20%3D-2-%5Cleft(%20-1%20%5Cright)%20%3D%5Ccolor%7Bred%7D%7B-1%7D

%5Ccolor%7Bred%7D%7Bf%5Cleft(%205%20%5Cright)%20%3D%7Df%5Cleft(%204%20%5Cright)%20-f%5Cleft(%203%20%5Cright)%20%3D-1-%5Cleft(%20-2%20%5Cright)%20%3D%5Ccolor%7Bred%7D%7B1%7D

%5Ccolor%7Bred%7D%7Bf%5Cleft(%206%20%5Cright)%20%3D%7Df%5Cleft(%205%20%5Cright)%20-f%5Cleft(%204%20%5Cright)%20%3D1-%5Cleft(%20-1%20%5Cright)%20%3D%5Ccolor%7Bred%7D%7B2%7D

%5Ccolor%7Bred%7D%7Bf%5Cleft(%207%20%5Cright)%3D%7D%20f%5Cleft(%206%20%5Cright)%20-f%5Cleft(%205%20%5Cright)%20%3D2-1%3D%5Ccolor%7Bred%7D%7B1%3Df%5Cleft(%201%20%5Cright)%20%7D

所以f%5Cleft(%20x%20%5Cright)%20是以%5Ccolor%7Bred%7D%7B6%7D最小正周期的函數(shù).

所以

%5Cbegin%7Baligned%7D%0A%09%5Csum_%7Bk%3D1%7D%5E%7B22%7D%7Bf%5Cleft(%20k%20%5Cright)%7D%26%3D3%5Csum_%7Bk%3D1%7D%5E6%7Bf%5Cleft(%20k%20%5Cright)%7D%2B%5Csum_%7Bk%3D1%7D%5E4%7Bf%5Cleft(%20k%20%5Cright)%7D%5C%5C%0A%09%26%3D3%5Ctimes%200%2B1-1-2-1%5C%5C%0A%09%26%3D-3%5C%5C%0A%5Cend%7Baligned%7D

A.


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